advanced equity derivatives volatility and correlations

Advanced EquityDerivatives

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Volatility and Correlation

SÉBASTIEN BOSSU

Cover image: MGB (artmgb.com)Cover design: Wiley

Copyright © 2014 by Sébastien Bossu. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Bossu, Sébastien.Advanced equity derivatives : volatility and correlation / Sébastien Bossu.

pages cm. – (Wiley finance series)Includes bibliographical references and index.ISBN 978-1-118-75096-4 (cloth); ISBN 978-1-118-77484-7 (ePDF);

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“As for the expense,” gravely declaredthe deputy Haffner who never opened his mouth

except on great occasions, “our children will pay for it,and nothing will be more just.”

Emile Zola, La Curée (The Kill)

Contents

Foreword xi

Preface xiii

Acknowledgments xv

CHAPTER 1Exotic Derivatives 1

1-1 Single-Asset Exotics 11-2 Multi-Asset Exotics 41-3 Structured Products 9

References 11Problems 11

CHAPTER 2The Implied Volatility Surface 15

2-1 The Implied Volatility Smile and Its Consequences 152-2 Interpolation and Extrapolation 202-3 Implied Volatility Surface Properties 222-4 Implied Volatility Surface Models 22

References and Bibliography 29Problems 30

CHAPTER 3Implied Distributions 33

3-1 Butterfly Spreads and the Implied Distribution 333-2 European Payoff Pricing and Replication 363-3 Pricing Methods for European Payoffs 393-4 Greeks 41


vii

viii CONTENTS

CHAPTER 4Local Volatility and Beyond 45

4-1 Local Volatility Trees 454-2 Local Volatility in Continuous Time 464-3 Calculating Local Volatilities 484-4 Stochastic Volatility 50


CHAPTER 5Volatility Derivatives 59

5-1 Volatility Trading 595-2 Variance Swaps 615-3 Realized Volatility Derivatives 655-4 Implied Volatility Derivatives 67


CHAPTER 6Introducing Correlation 73

6-1 Measuring Correlation 736-2 Correlation Matrices 756-3 Correlation Average 776-4 Black-Scholes with Constant Correlation 826-5 Local Volatility with Constant Correlation 84


CHAPTER 7Correlation Trading 87

7-1 Dispersion Trading 877-2 Correlation Swaps 91

Problems 93

CHAPTER 8Local Correlation 95

8-1 The Implied Correlation Smile and Its Consequences 958-2 Local Volatility with Local Correlation 978-3 Dynamic Local Correlation Models 998-4 Limitations 99


Contents ix

CHAPTER 9Stochastic Correlation 103

9-1 Stochastic Single Correlation 1039-2 Stochastic Average Correlation 1049-3 Stochastic Correlation Matrix 108


Appendix A Probability Review 115A-1 Standard Probability Theory 115A-2 Random Variables, Distribution, and Independence 116A-3 Conditioning 117A-4 Random Processes and Stochastic Calculus 118

Appendix B Linear Algebra Review 119B-1 Euclidean Spaces 119B-2 Square Matrix Decompositions 120

Solutions Manual 123

Author’s Note 143

About the Author 145

Index 147

Foreword

Iam pleased to introduce Sébastien Bossu’s new book, Advanced EquityDerivatives, which is a great contribution to the literature in our field. Years

of practical experience as an exotics structurer, combined with strong theo-retical skills, allowed Sébastien to write a condensed yet profound text on avariety of advanced topics: volatility derivatives and volatility trading, cor-relation modeling, dispersion trading, local and stochastic volatility models,to name just a few.

This book not only reviews the most important concepts and recentdevelopments in option pricing and modeling, but also offers insightfulexplications of great relevance to researchers as well as traders. For instance,readers will find formulas to overhedge convex payoffs, the derivation ofFeller conditions for the Heston model, or an exposition of the latest localcorrelation models to correctly price basket options.

Perhaps the most exciting aspect of this book is its treatment of the latestgeneration of equity derivatives, namely volatility and correlation deriva-tives. Readers will find a wealth of information on these new securities,including original analyses and models to approach their valuation. Thechapters on correlation are particularly commendable, as they shed light onan otherwise still obscure area.

The content quality, selection of topics, and level of insight truly set thisbook apart. I have no doubt that equity derivatives practitioners around theworld, be they traders, quants or investors, will find it extremely pertinent,and I wish this book every success.

Peter Carr

Dr. Peter Carr has over 18 years of experience in the derivatives industryand is currently Global Head of Market Modeling at Morgan Stanley, as wellas Executive Director of the Math Finance program at NYU’s Courant Insti-tute. He has over 70 publications in academic and industry-oriented journalsand serves as an associate editor for eight journals related to mathemati-cal finance. Dr. Carr is also the Treasurer of the Bachelier Finance Society,a trustee for the Museum of Mathematics in New York, and has receivednumerous awards, including Quant of the Year by Risk magazine in 2003,the ISA Medal for Science in 2008, and Financial Engineer of the Year in2010.

xi

Preface

In 2004, while working as an equity derivatives analyst at J.P. Morgan inLondon, I came upon an esoteric trade: someone was simultaneously sell-

ing correlation and buying it back for a (risky) profit using two differentmethods. I became obsessed with the rationale behind this trade, and, afterwriting down the math, I discovered with excitement that with some cor-rections this trade led to a pure dynamic arbitrage strategy—the kind younormally find only in textbooks.

I could see, however, that transaction costs and other market frictionsmade the strategy very hard to implement in practice, especially for pricetakers on the buy side. But the fact remained that correlation could be boughtand sold at very different prices, and that didn’t make sense to me. So Ideveloped a simple “toy” model to see how this gap might be accounted for,and as I suspected I found that there should be little difference. What thismeant is that one of the two correlation instruments involved in the trade,namely the correlation swap, was not priced at “fair value” according to myanalysis.

Later on I refined my model, which I introduce in the last chapter of thisbook among other topics, and reached similar conclusions. I am very pleasedthat the topic of equity correlation has gained tremendous momentum since2004, and it is one of this book’s ambitions to introduce the work of othersin this highly specialized field. I have no doubt that many new exciting resultsare yet to be discovered in the coming years.

I also wanted to cover other key advanced concepts in equity derivativesthat are relevant to traders, quantitative analysts, and other professionals.Many of these concepts, such as implied distributions and local volatilities,are now well-known and established in the field, while others, such as localand stochastic correlation, lie at the forefront of current research.

To get the most out of this book, readers must already be familiar withthe terminology and standard pricing theory of equity derivatives, whichcan be found in my textbook Introduction to Equity Derivatives: Theory &Practice, second edition, also published by John Wiley & Sons.

I relied on a fair amount of advanced mathematics, and therefore a grad-uate scientific education is a prerequisite here, especially for those readerswho want to solve the problems included at the end of each chapter.

xiii

xiv PREFACE

The book is made of nine chapters, which are meant to be read sequen-tially, starting with an exposition of the most widespread exotic derivativesand culminating with cutting-edge concepts on stochastic correlation, whichare necessary to correctly price the next generation of equity derivatives suchas correlation swaps.

Some simplifications, such as zero interest rates and dividends, wereoften necessary to avoid convoluted mathematical expressions. I stronglyencourage readers to check the particular assumptions used for each formulabefore transposing it into another context.

I hope this book will prove insightful and useful to its target audience.I am always interested to hear feedback; please do not hesitate to contact meto share your thoughts.

Acknowledgments

Iwould like to thank Peter Carr for his foreword, and David Hait and histeam at OptionMetrics for providing me with very useful option data. I am

grateful to my team atWiley—Bill Falloon, Meg Freeborn, and associates—for their guidance and professionalism throughout the publication process.

Many thanks also to a group of individuals who, directly or indirectly,made this book possible: Romain Barc, Martin Bertsch, Eynour Boutia,Jose Casino, Mauro Cesa, John Dattorro, Emanuel Derman, Jim Gatheral,Fabrice Rouah, Simone Russo, Roberto Silvotti, and Paul Wilmott.

Last, a special mention to John Lyttle at Ogee Group for his help onmany figures and problem solutions.

xv

CHAPTER 1Exotic Derivatives

Strictly speaking, an exotic derivative is any derivative that is not a plainvanilla call or put. In this chapter we review the payoff and properties of themost widespread equity derivative exotics.

1-1 SINGLE-ASSET EXOTICS

1-1.1 Digital Options

A European digital or binary option pays off $1 if the underlying asset priceis above the strike K at maturity T, and 0 otherwise:

Digital Payoff =

{1 if ST > K0 otherwise

In its American version, which is more uncommon, the option pays off$1 as soon as the strike level is hit.

The Black-Scholes price formula for a digital option is simply:

e−rTN(d2) = e−rTN

(ln (F∕K) − 1

2𝜎2T

𝜎√T

)

where F is the forward price of S for maturity T, r is the continuous interestrate, and 𝜎 is the volatility parameter. When there is an implied volatilitysmile this formula is inaccurate and a corrective term must be added (seeSection 2-1.3).

Digital options are not easy to dynamically hedge because their deltacan become very large near maturity. Exotic traders tend to overhedge them

1

2 ADVANCED EQUITY DERIVATIVES

with a tight call spread whose range may be determined according to severalpossible empirical rules, such as:

■ Daily volatility rule: Set the range to match a typical stock price moveover one day. For example, if the annual volatility of the underlyingstock is 32% annually; that is, 32%/

√252 ≈ 2% daily, a digital option

struck at $100 would be overhedged with $98–$100 call spreads.■ Normalized liquidity rule: Set the range so that the quantity of callspreads is in line with the market liquidity of call spreads with 5%range. The quantity of call spreads is N/R where N is the quantity ofdigitals and R is the call spread range. If the tradable quantity of callspreads with range 5% is V, the normalized tradable quantity ofcall spreads with range R would be V × R / 0.05. Solving for R gives

R =√

0.05 × NV. In practice V is either provided by the option trader

or estimated using the daily trading volume of the stock.

1-1.2 Asian Options

In an Asian call or put, the final underlying asset price is replaced by anaverage:

Asian Call Payoff = max(0, AT −K)Asian Put Payoff = max(0, K−AT)

where AT = 1n

n∑i=1

Sti for a set of pre-agreed fixing dates t1 < t2 < · · · < tn ≤ T.

For example, a one-year at-the-money Asian call on the S&P 500 index withquarterly fixings pays off max

(0, S0.25+S0.5+S0.75+S1

4− S0

), where S0 is the cur-

rent spot price and S0.25,… , S1 are the future spot prices observed every threemonths.

On occasion, the strike may also be replaced by an average, typicallyover a short initial observation period.

Fixed-strike Asian options are always cheaper than their Europeancounterparts, because AT is less volatile than ST.

There is no closed-form Black-Scholes formula for arithmeticAsian options. However, for geometric Asian options where AT =exp

[1T∫ T0 ln St dt

], the Black-Scholes formulas may be used with adjusted

volatility ⌢𝜎 = 𝜎∕√3 and dividend yield ⌢q = 12

(r + q + 𝜎

2

6

), as shown in

Problem 1.3.A common numerical approximation for the price of arithmetic Asian

options is obtained by fitting a lognormal distribution to the actual risk-neutral moments of AT.

Exotic Derivatives 3

1-1.3 Barrier Options

In a barrier call or put, the underlying asset price must hit, or never hit, acertain barrier level H before maturity:

■ For a knock-in option, the underlying must hit the barrier, or else theoption pays nothing.

■ For a knock-out option, the underlying must never hit the barrier, orelse the option pays nothing.

Barrier options are always cheaper than their European counterparts,because their payoff is subject to an additional constraint. On occasion, afixed cash “rebate” is paid out if the barrier condition is not met.

Similar to digital options, barrier options are not easy to dynamicallyhedge: their delta can become very large near the barrier level. Exotic traderstend to overhedge them by shifting the barrier a little in their valuationmodel.

Continuouslymonitored barrier options have closed-formBlack-Scholesformulas, which can be found, for instance, in Hull (2012). The preferredpricing approach is the local volatility model (see Chapter 4).

In practice the barrier is often monitored on a set of pre-agreed fixingdates t1 < t2 < · · · < tn ≤ T. Monte Carlo simulations are then commonlyused for valuation.

Broadie, Glasserman, and Kou (1997) derived a nice result to switchbetween continuous and discrete barrier monitoring by shifting the barrierlevel H by a factor exp (±𝛽𝜎

√Δt) where 𝛽 ≈ 0.5826, σ is the underlying

volatility, and Δt is the time between two fixing dates.

1-1.4 Lookback Options

A lookback call or put is an option on the maximum or minimum pricereached by the underlying asset until maturity:

Lookback call payoff = max (0, max0≤t≤T St − K);

Lookback put payoff = max (0,K − min0≤t≤T St).

Lookback options are always more expensive than their European coun-terparts: about twice as much when the strike is nearly at the money, asshown in Problem 1.5.

Continuously monitored lookback options have closed-form Black-Scholes formulas, which can be found, for instance, in Hull (2012). Thepreferred pricing approach is the local volatility model (see Chapter 4).

In practice the maximum or minimum is often monitored on a set ofpre-agreed fixing dates t1 < t2 < · · · < tn ≤ T. Monte Carlo simulations arethen commonly used for valuation.


1-1.5 Forward Start Options

In a forward start option the strike is determined as a percentage k of thespot price on a future start date t0 > 0:

Forward start call payoff = max(0, ST − kSt0 );

Forward start put payoff = max(0,kSt0 − ST).

At t = t0 a forward start option becomes a regular option. Note that theforward start feature is not specific to vanilla options and can be added toany exotic option that has a strike.

Forward start options have closed-form Black-Scholes formulas. Thepreferred pricing approach is to use a stochastic volatility model (seeChapter 4).

1-1.6 Cliquet Options

A cliquet or ratchet option consists of a series of consecutive forward startoptions, for example:

Monthly cliquet option payoff = max

[0,

12∑i=1

min

(5%,

Si∕12S(i−1)∕12

− 1

)]where 5% is the local cap amount. In other words, this particular cliquetoption pays off the greater of zero and the sum of monthly returns, eachcapped at 5%.

Cliquet options can be very difficult to value and especially hedge.

1-2 MULTI-ASSET EXOTICS

Multi-asset exotics are based on several underlying stocks or indices, andthus their fair value depends on the level of correlation between the under-lying assets. They are typically priced on a Monte Carlo simulation enginewith local volatilities (see Chapter 4 and Chapter 6, Section 6-5).

1-2.1 Spread Options

The payoff of a spread option is based on the difference in gross returnbetween two underlying assets:

Spread option payoff = max

(0,S(1)T

S(1)0

−S(2)T

S(2)0

− k

)


where k is the residual strike level (in %). For example, a spread option onApple Inc. vs Google Inc. with 5% strike pays off the outperformance ofApple over Google in excess of 5%: if Apple’s return is 13% and Google’sis 4%, the option pays off 13%− 4%−5% = 4%.

The value of a spread option is very sensitive to the level of correlationbetween the two assets. Specifically the option value increases as correlationdecreases: the lower the correlation, the wider the two assets are expectedto spread apart.

In practice hedging spread options can be difficult because the spreadS(1)T

S(1)0

−S(2)T

S(2)0

is often nearly orthogonal to the basket 12

[S(1)T

S(1)0

+S(2)T

S(2)0

]− 1.

When k = 0 a spread option is also known as an exchange option. Aclosed-form Black-Scholes formula is then available which can be found, forinstance, in Hull (2012).

1-2.2 Basket Options

A basket call or put is an option on the gross return of a portfolio of nunderlying assets:

Basket call payoff = max

(0,

n∑i=1

wi

S(i)T

S(i)0− k

);

Basket put payoff = max

(0,k −

n∑i=1

wi

S(i)T

S(i)0

),

where the weights w1,… , wn sum to 100% and the strike k is expressed asa percentage (e.g., 100% for at the money).

EXAMPLE

Equally-Weighted Stock Basket Call

Option seller: ABC Bank Co.Notional amount: $20,000,000Issue date: [Today]Maturity date: [Today + 3 years]

(Continued)


EXAMPLE (Continued)

Equally-Weighted Stock Basket Call

Underlying stocks: IBM (IBM), Microsoft (MSFT), Google (GOOG)Payoff:

Notional ×max(0, 1

3

(IBMfinal

IBMinitial+

MSFTfinal

MSFTinitial+

GOOGfinal

GOOGinitial

)− 1

)Option price: 17.4%

The value of a basket option is sensitive to the level of pairwise corre-lations between the assets. The lower the correlation, the less volatile theportfolio and the cheaper the basket option.

Basket options do not have closed-form Black-Scholes formulas. A com-mon approximation technique is to fit a lognormal distribution to the actualmoments of the basket and then use formulas for the single-asset case.

1-2.3 Worst-Of and Best-Of Options

A worst-of call or put is an option on the lowest gross return between nunderlying assets:

Worst-of call payoff = max

(0, min

1≤i≤nS(i)T

S(i)0− k

);

Worst-of put payoff = max

(0,k − min

1≤i≤nS(i)T

S(i)0

),

where the strike k is expressed as a percentage (e.g., 100% for at the money).For example, a worst-of at-the-money call on Apple, Google, and Microsoftpays off the worst stock return between the three companies, if positive.

Similarly, a best-of call or put is an option on the highest gross returnbetween n underlying assets.

Worst-of calls and best-of puts are always cheaper than any of theirsingle-asset European counterparts, while best-of calls and worst-of puts arealways more expensive.


1-2.4 Quanto Options

The payoff of a quanto option is paid out in a different currency from theunderlying assets, at a guaranteed exchange rate. For example, a call on theS&P 500 index quanto euro pays off max(0, ST −K) in euros instead ofdollars, thereby guaranteeing an exchange rate of 1 euro per dollar.

The actual exchange rate between the asset currency and the quanto cur-rency is in fact an implicit additional underlying asset. The value of quantooptions is very sensitive to the correlation between the primary asset and theimplicit exchange rate.

Quanto options are an example of hybrid exotic options involving dif-ferent asset classes—here equity and foreign exchange.

In terms of pricing, the quanto feature is often approached using a tech-nique called change of numeraire. In summary, this technique says that therisk-neutral dynamics of an asset quantoed in a different currency from itsnatural currency has the same volatility coefficient but an adjusted driftcoefficient.

FOCUS ON CHANGE OF NUMERAIRE

This technique builds upon the concepts of change of measure andGirsanov’s theorem, which are explained in Appendix 1.A.

Consider a world with two currencies, say dollars and euros, anda non-income-paying asset S with dollar price S$ and euro price S€.DenoteX the exchange rate of one dollar into euros, so that S€t = S$t Xt.Assume that S$ andX both follow a geometric Brownian motion underthe dollar risk-neutral measure ℚ$, specifically:

For S$∶ dS$t ∕S$t = r$dt + 𝜎dWt

For X∶ dXt∕Xt = 𝜈dt + 𝜂dZt

whereW,Z are standard Brownian motions underℚ$ with correlation𝜌, r$ is the dollar interest rate, and all other parameters are free.

Because the original Girsanov theorem applies to independentBrownianmotions, we rewriteZ = 𝜌W + 𝜌W⟂ whereW⟂ is a standardBrownian motion under ℚ$ independent from W and 𝜌 =

√1 − 𝜌2 is

the orthogonal complement of 𝜌. The diffusion equation for X thenbecomes:

dXt∕Xt = 𝜈dt + 𝜂𝜌dWt + 𝜂𝜌dW⟂t


Applying the Ito-Doeblin theorem to the product S€t = S$t Xt weobtain after simplifying:

dS€t ∕S€t = (r$ + 𝜈 + 𝜌𝜎𝜂)dt + (𝜎 + 𝜌𝜂)dWt + 𝜂𝜌dW⟂

t (1.1)

Because S€ is a euro tradable asset we must also have:

dS€t ∕S€t = r€dt + (𝜎 + 𝜌𝜂)dW̃t + 𝜂𝜌dW̃⟂

t (1.2)

where W̃, W̃⟂ are independent standard Brownian motions under theeuro risk-neutral measureℚ€. This is the diffusion equation of the com-posite asset S€ after conversion from dollars to euros.

The processes W̃, W̃⟂ are affine transformations of the originalprocesses W,W⟂; specifically:{

W̃t = Wt + 𝛾1tW̃⟂

t = W⟂t + 𝛾2t

where 𝛾1 and 𝛾2 are particular coefficients. Substituting into Equation(1.1) and connecting with Equation (1.2) we obtain that 𝛾1, 𝛾2 mustsatisfy:

r$ + 𝜈 + 𝜌𝜎𝜂 = r€ + 𝛾1(𝜎 + 𝜌𝜂) + 𝛾2𝜂𝜌

In order to determine 𝛾1, 𝛾2 uniquely, we need another equation.This is provided by the dynamics of X, which is a euro-tradable asset(it is the price in euros of $1):

dXt∕Xt = (r€ − r$)dt + 𝜂𝜌dW̃t + 𝜂𝜌dW̃⟂t

Following the same reasoning we find that 𝛾1, 𝛾2 must also satisfy:

𝜈 = r€ − r$ + 𝛾1𝜂𝜌 + 𝛾2𝜂𝜌

Solving for 𝛾1, 𝛾2 we find:

⎧⎪⎨⎪⎩𝛾1 = 𝜂𝜌

𝛾2 =𝜈 + r$ − r€ − 𝜂2𝜌2

𝜂𝜌

The dynamics of S$ may thus be rewritten as:

dS$t ∕S$t = r$dt + 𝜎dWt = r$dt + 𝜎(dW̃t − 𝛾1dt) = (r$ − 𝜌𝜎𝜂)dt + 𝜎dW̃t


This is the diffusion equation for S$ quanto euro. In particular, theforward price of S$ quanto euro is:

𝔼ℚ€(S$T) = S$0e

(r$−𝜌𝜎𝜂)T

1-3 STRUCTURED PRODUCTS

Structured products combine several securities together, especially exoticoptions. They are typically sold as equity-linked notes (ELN) or mutualfunds to small investors as well as large institutions. These notes and fundsare sometimes traded on exchanges.

EXAMPLES

Capital GuaranteedPerformance Note Reverse Convertible Note

Issuer: ABC Bank Co. Issuer: ABC Bank Co.Notional amount: $10,000,000 Notional amount: €2,000,000Issue date: [Today] Issue date: [Today]Maturity date: [Today + 5 years] Maturity date: [Today + 3 years]Underlying index: S&P 500 (SPX) Underlying stock: Kroger Co.

(KR)Payoff:

Notional ×[100%+Participation

×max(0,

SPXfinal

SPXinitial− 1

)]Payoff:

(a) If, between the start and matu-rity dates, Kroger Co. alwaystrades above the Barrier level,Issuer will pay:

Notional ×max(115%,

SfinalSinitial

)(b) Otherwise, Issuer will pay:

Notional ×SfinalSinitial

Participation: 50% Barrier level: 70%


In the Capital Guaranteed Performance Note, investors are guaranteed1

to get their $10mn capital back after five years. This is much safer than adirect $10mn investment in the S&P 500 index, which could result in a loss.In exchange for this protection, investors receive a smaller share in the S&P500 performance: 50% instead of 100%.

In the Reverse Convertible Note, investors may lose on their €2mn cap-ital if Kroger Co. ever trades below the 70% barrier, but never more thana direct investment in the stock (ignoring dividends). Otherwise, investorsreceive at least €2.3mn after three years, and never less than a direct invest-ment in the stock (again, ignoring dividends).

In some cases it is possible to break down a structured product intoa portfolio of securities whose prices are known and find its value. In allother cases the payoff is typically programmed on aMonte Carlo simulationengine.

Multi-asset structured products significantly expand the payoff possi-bilities of exotic options. They allow investors to play on correlation andexpress complex investment views.

EXAMPLE

Worst-Of Reverse Convertible Note Quanto CHF

Issuer: ABC Bank Co.Notional amount: CHF 5,000,000Issue date: [Today]Maturity date: [Today + 3 years]Underlying indexes: S&P 500 (SPX), EuroStoxx-50 (SX5E), Nikkei225 (NKY)

Payoff:

(a) If, between the start and maturity dates, all underlying indexesalways trade above the Barrier level, Issuer will pay:

Notional ×max(120%,min

(SPXfinal

SPXinitial,SX5Efinal

SX5Einitial,NKYfinal

NKYinitial

))(Continued)

1Provided the issuer does not go bankrupt.


EXAMPLE (Continued)

Worst-Of Reverse Convertible Note Quanto CHF

(b) Otherwise, Issuer will pay:

Notional ×min(SPXfinal

SPXinitial,SX5Efinal

SX5Einitial,NKYfinal

NKYinitial

)Barrier level: 50% of Initial Price

Multi-asset structured product valuation is almost always done usingMonte Carlo simulations. Hedging correlation risk is often difficult orexpensive, and exotic trading desks tend to accumulate large exposures,which can cause significant losses during a market crash.

REFERENCES

Baxter, Martin, and Andrew Rennie. 1996. Financial Calculus: An Introduction toDerivative Pricing. New York: Cambridge University Press.

Broadie, Mark, Paul Glasserman, and Steven Kou. 1997. “A Continuity Correctionfor Discrete Barrier Options.” Mathematical Finance 7 (4): 325–348.

Hull, John C. 2012. Option, Futures, and Other Derivatives, 8th ed. New York:Prentice Hall.

PROBLEMS

1.1 “Free” Option

Consider a European call option on an underlying asset S with strike Kand maturity T where “you only pay the premium if you win,” that is,if ST > K.

(a) Draw the diagram of the net P&L of this “free” option at maturity. Is itreally “free”?

(b) Find a replicating portfolio for the “free” option using vanilla and exoticoptions.

(c) Calculate the fair value of the “free” option premium using the Black-Scholes model with 20% volatility, S0 = K = $100, one-year maturity,zero interest and dividend rates.


1.2 Autocallable

Consider an exotic option expiring in one, two, or three years on an under-lying asset S with the following payoff mechanism:

■ If after one year S1 > S0 the option pays off 1 + C and terminates;■ Else if after two years S2 > S0 the option pays off 1 + 2C and terminates;■ Else if after three years S3 > 0.7 × S0 the option pays off max(1 + 3C,S3∕S0);

■ Otherwise, the option pays off S3/S0.

Assuming S0 = $100, zero interest and dividend rates, and 25% volatil-ity, estimate the level of C so that the option is worth 1 using Monte Carlosimulations.

1.3 Geometric Asian Option

Consider a geometric Asian option on an underlying S with payoff f(AT)

where AT = exp(

1T∫ T0 ln St dt

). Assume that S follows a geometric Brown-

ian motion with parameters (r−q, 𝜎) under the risk-neutral measure.

(a) Using the Ito-Doeblin theorem, show that

AT = S0 exp

(12

(r − q − 1

2𝜎2

)T + 𝜎

T∫T

0Wtdt

)

(b) Using the Ito-Doeblin theorem, show that ∫ T0 Wtdt = ∫ T

0 (T − t)dWt.What is the distribution of this quantity?

(c) Show that AT is lognormally distributed with parameters(ln S0 +(

r − ⌢q − 12⌢𝜎2)T,⌢𝜎√T

)where ⌢𝜎 = 𝜎∕√3 and ⌢q = 1

2

(r + q + 𝜎

2

6

).

1.4 Change of Measure

In the context of Appendix 1.A, verify that 𝔼ℚ(ST) = S0erT using the expres-

sion for dℚ/dℙ.

1.5 At-the-Money Lookback Options

The Black-Scholes closed-form formula for an at-the-money lookback callis given as:

Lookback0 = e−rTS0(N(−𝛼2) − 1) + S0N(𝛼1)(1 + 𝜎

2

2r

)− e−rT𝜎2

2rS0N(𝛼3)

where 𝛼1,2 =(r𝜎± 1

2𝜎)√

T and 𝛼3 =(− r𝜎+ 1

2𝜎)√

T.


Using a first-order Taylor expansion of the cumulative normal distribu-tion N(⋅) show that for reasonable rates and maturities we have the proxy:

Lookback0 ≈4S0𝜎

√T√

2𝜋

which is twice as much as the European call proxy: c0 ≈ 2S0𝜎√T√

2𝜋.

1.6 Siegel’s Paradox

This problem is about foreign exchange rates and goes beyond the scope ofequity derivatives.

Consider two currencies, say dollars and euros, and suppose thattheir corresponding interest rates, r$ and r€, are constant. Let X be theeuro–dollar exchange rate defined as the number of dollars per euro. Thetraditional risk-neutral process for X is thus:

dXt = (r$ − r€)Xtdt + 𝜎XtdWt

where W is a standard Brownian motion.

(a) Using the Ito-Doeblin theorem, show that the risk-neutral dynamics forthe dollar-euro exchange rate, that is, the number 1/X of euros per dol-lar, is:

d1Xt

= (r€ − r$ + 𝜎2)1Xt

dt + 𝜎 1Xt

dWt

(b) Symmetry suggests that the drift of 1/X should be r€ − r$ instead—thisis Siegel’s paradox. Use your knowledge of quantos (see Section 1-2.4)to resolve the paradox.

APPENDIX 1.A: CHANGE OF MEASUREAND GIRSANOV’S THEOREM

Recall that the Black-Scholes model assumes that the underlying asset priceprocess follows a geometric Brownian motion:

dSt∕St = 𝜇dt + 𝜎dWt

where W is a standard Brownian motion under some objective probabilitymeasure ℙ, 𝜇 is the objective drift coefficient, and 𝜎 is the objective volatilitycoefficient.

However, the drift coefficient 𝜇 disappears from option pricingequations as a result of delta-hedging, and option prices may equivalentlybe calculated as discounted expected payoffs under a special probability


measure ℚ called risk-neutral. Under ℚ, the underlying asset price processfollows the geometric Brownian motion:

dSt∕St = rdt + 𝜎dW′t

whereW′ is a standard Brownian motion under ℚ, r is the continuous inter-est rate, and 𝜎 is the same volatility coefficient.

To understand how ℙ and ℚ relate, consider the undiscounted expectedpayoff:

𝔼ℚ(f (ST)) = ∫∞

0f (s)ℚ{ST = s}ds

= ∫∞

0f (s)

ℚ{ST = s}ℙ{ST = s}

ℙ{ST = s}ds

If we define the ratio of densities h(s) = ℚ{ST=s}ℙ{ST=s}

then we can write:

𝔼ℚ(f (ST)) = ∫∞

0f (s)h(s)ℙ{ST = s}ds = 𝔼ℙ(f (ST)h(ST)).

The change of measure from ℙ to ℚ is thus equivalent to multiplying bythe random variable h(ST) called a Radon-Nikodym derivative and properlydenoted dℚ

dℙ . Girsanov’s theorem states that ℚ exists and is properly definedby a Radon-Nikodym derivative of the form:

dℚdℙ

= exp(r − 𝜇𝜎

WT − 12

( r − 𝜇𝜎

)2T)

Furthermore, W′t = Wt +

r−𝜇𝜎t is then a Brownian motion under ℚ.

Problem 1.4 verifies that 𝔼ℚ(ST) = S0erT .

For a rigorous yet accessible exposition of the change of measure tech-nique and Girsanov’s theorem we refer the reader to Baxter and Rennie(1996).

CHAPTER 2The Implied Volatility Surface

Despite its flaws and limitations, the Black-Scholes model became the bench-mark to interpret option prices. Specifically, option prices are reverse-engineered to calculate implied volatilities, the same way that bond pricesare transformed into yields, which are easier to understand. This process,combined with interpolation and extrapolation techniques, gives rise to anentire surface along the strike and maturity dimensions.

2-1 THE IMPLIED VOLATILITY SMILEAND ITS CONSEQUENCES

The Black-Scholes model assumes a single constant volatility parameter toprice options. In practice, however, every listed vanilla option has a differentimplied volatility 𝜎∗(K,T) for each strikeK andmaturityT. Figure 2.1 showswhat an implied volatility surface (K,T) → 𝜎∗(K,T) looks like.

For a fixed maturity T the curve K → 𝜎∗(K,T) is called the impliedvolatility smile or skew and exhibits a downward-sloping shape, as shownin Figure 2.2. Note that in other asset classes, such as interest rates or cur-rencies, the smile tends to be symmetric rather than downward-sloping.

2-1.1 Consequence for the Pricing of Calland Put Spreads

A direct consequence of the implied volatility smile is that the Black-Scholesmodel gives inaccurate call spread and put spread prices. To illustrate thispoint, Figure 2.3 shows the Black-Scholes price of a one-year call spread withstrikes $100 and $110 as a function of the single Black-Scholes volatility

15


10

0.0

0.1

0.2

0.3

0.4

80

100

120

2

3

4

5

FIGURE 2.1 Implied volatility surface of the S&P 500 as of July 18, 2012. Strikesare in percentage of the spot level.

0%

10%

20%

30%

40%

50%

60%

70%

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Imp

lied

vo

latilit

y

Moneyness (Strike/Spot)

FIGURE 2.2 Implied volatility smile of S&P 500 index options expiring December20, 2014, as of July 22, 2012.Data source: Bloomberg.

parameter 𝜎. We can see that the curve peaks at 𝜎 ≈ 30.8% for a maximumprice of $3.78. Thus, no single value of 𝜎 may reproduce any market priceabove $3.78. Interestingly, this phenomenon is not symmetric: $90–$100put spreads can be priced with a single volatility parameter, but the value of𝜎 will be significantly off the level of implied volatility for each put.

The Implied Volatility Surface 17

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2

4

6

8

10C

all

spre

ad v

alu

e

Volatility σ

FIGURE 2.3 Black-Scholes value of a call spread as a function of volatility. Spotprice $100, one-year maturity, strikes $100 and $110, zero rates and dividends.

2-1.2 Consequence for Hedge Ratios

The smile does not mean that Black-Scholes is wrong and should be rejected.Practitioners typically price over-the-counter (OTC) vanilla options usingBlack-Scholes and an appropriate volatility interpolation or extrapolationscheme, for the simple reason that implied volatilities are derived from listedoption prices in the first place. In other words, the fact that the Black-Scholesmodel may be faulty is fairly irrelevant for vanilla option pricing.

However, the vanilla hedge ratios or Greeks are model-dependent andshould be adjusted for the smile. To see this, notice first how a change inspot price impacts the moneyness of an option: after a $1 uptick from aninitial $100 underlying spot price, an out-of-the-money call struck at $110is now only $9 out of the money. In the presence of the smile, we may wantto use a different implied volatility to reprice the call, and we must make anassumption on the behavior of the smile curve:

■ If we assume that the smile curve does not change at all, we shoulduse the same implied volatility to reprice the call. This is known as thesticky-strike rule and produces the same delta as Black-Scholes.

■ If we assume that the smile curve does not change with respect to mon-eyness (strike over spot), we should use the implied volatility of the110/101 × 100 ≈ $108.91 call to reprice the call. This is known as thesticky-moneyness rule and produces a higher delta than Black-Scholes.


■ If we assume that the smile curve does not change with respect to delta,we should use the implied volatility corresponding to the new delta toreprice the call. This is known as the sticky-delta rule and produces ahigher delta than Black-Scholes. Note that the consistent definition ofdelta is circular in this case.

Other rules would obviously produce different results.

FOCUS ON THE DELTA GENERATED BY THE SMILE

Assuming that the smile 𝜎∗(S,K,T) depends on spot, strike, and matu-rity, and denoting cBS(S,K,T, r, 𝜎) the Black-Scholes formula for theEuropean call, we have by the chain rule:

𝛿 = ddScBS(S,K,T, r, 𝜎∗(S,K,T)) =

𝜕cBS𝜕S

+ 𝜕𝜎∗

𝜕S×𝜕cBS𝜕𝜎

= 𝛿BS +VBS ×𝜕𝜎∗

𝜕S

where 𝛿BS, VBS are the Black-Scholes delta and vega, respectively.

■ In the sticky-strike rule we have 𝜕𝜎∗

𝜕S= 0 and thus 𝛿 = 𝛿BS.

■ In the sticky-moneyness rule we may rewrite 𝜎∗(S,K,T) ≡𝜎∗1

(KS,T

)and thus 𝛿 = 𝛿BS − VBS

KS2𝜕𝜎∗

1𝜕S

(KS,T

).

■ To approach the sticky-delta rule we may rewrite 𝜎∗(S,K,T) ≡𝜎∗2

(ln K

S,T

)since the Black-Scholes delta is N(d1) =

N

(ln(S∕K)+

(r+ 1

2 𝜎2)T

𝜎√T

)which is a function of ln(K/S). Thus

𝛿 = 𝛿BS − VBS1S

𝜕𝜎∗2𝜕S

(ln K

S,T

).

2-1.3 Consequence for the Pricing of Exotics

In Chapter 3 we will see that European exotic payoffs can in theory be repli-cated by a static portfolio of vanilla options along a continuum of strikes. Inthe absence of arbitrage, the price of the exotic option must match the priceof the portfolio. Thus it would be inaccurate to use the Black-Scholes modelto price the exotic option in the presence of the smile.

As a fundamental example consider the digital option that pays off $1at maturity T if the final spot price ST is above the strike K, and 0 otherwise.


The Black-Scholes value for the digital option is simply:

DBS(S,K, r,T, 𝜎) = e−rTN(d2) = e−rTN

⎛⎜⎜⎜⎝ln (S∕K) +

(r − 1

2𝜎2

)T

𝜎√T

⎞⎟⎟⎟⎠where S is the spot price, r is the continuous interest rate for maturity T, and𝜎 is the constant Black-Scholes volatility parameter.

Digital options are difficult to delta-hedge because their delta becomesvery large around the strike as maturity approaches. Equity exotic traderswill typically overhedge them with tight call spreads. For example, tooverhedge a digital paying off $1,000,000 above a strike of $100 and0 otherwise, a trader might buy 200,000 call spreads with strikes $95and $100.

Figure 2.4 shows how in general a quantity 1/𝜀 of call spreads withstrikes K − 𝜀 and K will overhedge a digital option struck at K. In the limitas 𝜀 goes to zero, we obtain an exact hedging portfolio whose price is:

D(S,K, r,T) = DBS(S,K, r,T, 𝜎∗(K,T)) − VBS(S,K, r,T, 𝜎∗(K,T)) ×𝜕𝜎∗

𝜕K

where 𝜎*(K, T) is the implied volatility for strikeK and maturity T andVBS isthe Black-Scholes vega of a vanilla option. Because the equity smile is mostlydownward-sloping we typically have 𝜕𝜎

∗

𝜕K< 0 and thus the digital option is

worth more than its Black-Scholes value.

Payoff

$1

K – ε K

1/ε call spreads digital

Spot

FIGURE 2.4 Digital and leveraged call spreadpayoffs.


2-2 INTERPOLATION AND EXTRAPOLATION

Implied volatilities derived from listed option prices are only available for afinite number of listed strikes and maturities. However, on the OTC market,option investors will ask for quotes for any strike or maturity, and it is impor-tant to be able to interpolate or extrapolate implied volatilities.

Interpolation is relatively easy: for a given maturity, if the at-the-moneyoption has 20% implied volatility and the $90-strike option has 25%implied volatility, it intuitively makes sense to linearly interpolate and saythat the $95-strike option should have a 22.5% implied volatility. Similarly,for a given strike, we can linearly interpolate implied volatility through time.

One issue with linear interpolation, however, is that it produces acracked smile curve. More sophisticated interpolation techniques, such ascubic splines, are often used to obtain a smooth curve. Figure 2.5 comparesthe two methods.

It must be emphasized that unconstrained interpolation methods mayproduce arbitrageable volatility surfaces. Several papers listed in Homescu(2011) discuss how to eliminate arbitrage.

On the other hand extrapolation is a difficult endeavor: how to pricea five-year option if the longest listed maturity is two years? There is nodefinite answer to this question, and we must typically resort to a volatilitysurface model (see Section 2-4).

Note that extrapolating a cubic spline fit tends to produce unpredictableresults and should be avoided at all costs.

0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.6

0.5

0.4

0.3

0.2

0.1

Implie

d v

ola

tilit

y σ

*

Moneyness (Strike/Spot)

FIGURE 2.5 Comparison of linear (solid line) and cubic splines (dashed line)interpolation methods.


FOCUS ON CUBIC SPLINE INTERPOLATION

Given n + 1 points (xi, yi)0≤i≤n we want to find a smooth function S(x)that connects all the points. This may be done by defining n separatecubic polynomials for each interval [xi,xi+1] where i = 0, 1,… , n − 1of the form:

Si(x) = ai(x − xi)3 + bi(x − xi)2 + ci(x − xi) + di for xi ≤ x ≤ xi+1

The spline interpolation function is the piecewise combination ofthe n functions S0, S1,… , Sn–1:

S(x) =

⎧⎪⎪⎨⎪⎪⎩S0 (x) if x0 ≤ x ≤ x1S1(x) if x1 ≤ x ≤ x2⋮ ⋮

Sn−1(x) if xn−1 ≤ x ≤ xn

There are thus 4n coefficients (ai,bi, ci,di) to be chosen, for whichwe need 4n conditions. A first set of 2n conditions is provided by therequirement that S(x) should connect all the points; that is: Si(xi) = yiand Si(xi+1) = yi+1.

Another set of 2(n − 1) conditions is provided by requiring S(x) tohave continuous first- and second-order derivatives; that is: S′i(xi+1) =S′i+1(xi+1) and S

′′i (xi+1) = S′′i+1(xi+1) for i = 0, 1,… , n − 2.

This only leaves two additional conditions to be found in order touniquely determine all coefficients (ai,bi, ci,di). A common choice is tofurther require that S′′0 (x0) = 0 and S′′n−1(xn) = 0.

Assuming for ease of exposure that all intervals are of unit length(i.e., xi+1 − xi = 1), and rearranging all equations, we obtain b0 = 0and a beautiful tridiagonal linear system of the form:

⎡⎢⎢⎢⎢⎢⎢⎣

2 1 (0)1 4 1

1 4 ⋱1 ⋱ 1⋱ 4 1

(0) 1 2

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

b1b2b3⋮

bn−2bn−1

⎤⎥⎥⎥⎥⎥⎥⎦= 3

⎡⎢⎢⎢⎢⎢⎢⎣

y0 − 2y1 + y2y1 − 2y2 + y3y2 − 2y3 + y4

⋮yn−3 − 2yn−2 + yn−1yn−2 − 2yn−1 + yn

⎤⎥⎥⎥⎥⎥⎥⎦The computation of the other coefficients is then straightforward.


2-3 IMPLIED VOLATILITY SURFACE PROPERTIES

Not every surface f(K, T) is a candidate for an implied volatility surface𝜎∗(K,T). Denote c(S,K,T, r), p(S,K,T, r) the call and put values inducedby 𝜎∗(K,T), respectively. To preclude arbitrage we must at least require:

■ No call or put spread arbitrage∶ 𝜕c𝜕K

≤ 0,𝜕p𝜕K

≥ 0 (2.1)

■ No butterfly spread arbitrage∶ 𝜕2c𝜕K2

≥ 0,𝜕2p

𝜕K2≥ 0 (2.2)

■ No calendar spread arbitrage1∶ 𝜕c𝜕T

≥ 0,𝜕p𝜕T

≥ 0 (2.3)

These inequalities place upper and lower bounds on 𝜎∗(K,T) and itsderivatives. For example, by the chain rule applied to 𝜕c

𝜕K= 𝜕

𝜕KcBS(S,K,

T, r, 𝜎∗(K,T)), we obtain 𝜕c𝜕K

= 𝜕cBS𝜕K

+ 𝜕cBS𝜕𝜎

𝜕𝜎∗

𝜕K, and thus 𝜕c

𝜕K≤ 0 is equivalent

to the upper bound 𝜕𝜎∗

𝜕K≤ − 𝜕cBS∕𝜕K

𝜕cBS∕𝜕𝜎

||||𝜎=𝜎∗(K,T).When designing an implied volatility surface model, it is important to

check that these constraints are satisfied.The implied volatility surface must also satisfy certain asymptotic prop-

erties. Perhaps the most notable one for fixed maturity is that implied vari-ance, the square of implied volatility, is bounded from above by a functionlinear in log-strike as kF → 0 and kF → ∞:

𝜎∗2(kF,T) ≤ 𝛽T|lnkF|where kF = K/F is the forward-moneyness and β ∈ [0, 2] is different for eachlimit. This result is more rigorously expressed with supremum limits and werefer the interested reader to Lee (2004).

2-4 IMPLIED VOLATILITY SURFACE MODELS

Every large equity option house maintains several proprietary models of theimplied volatility surface, which are used by their market-makers to markpositions. By definition these models are not in the public domain, and wemust regrettably leave them in the dark. Fortunately some researchers havepublished their models and we now present a selection.

1For a given strike, American call and put prices must increase with maturity underpenalty of arbitrage. This requirement is frequently extended to European optionprices, although in theory small violations could be accepted.


There are two ways to model the volatility surface 𝜎∗(k,T):

1. Directly, by specifying a functional form such as a parametric function,or an interpolation and extrapolation method;

2. Indirectly, by modeling the behavior of the underlying asset differentlyfrom the geometric Brownian motion posited by Black-Scholes.

Here, it is worth distinguishing between two kinds of implied volatility𝜎∗: market-implied volatility 𝜎∗

Market, which is computed frommarket prices;

and model-implied volatility 𝜎∗[Model], which is induced by a volatility surfacemodel attempting to reproduce 𝜎∗

Market. However, for ease of notation we

will often keep the notation 𝜎* when there is no ambiguity nor need forsuch distinction.

2-4.1 A Parametric Model of Implied Volatility:The SVI Model

A popular parameterization of the smile for fixed maturity is the SVI modelby Gatheral (2004). SVI stands for stochastic volatility-inspired and has thesimple functional form:

𝜎∗SVI(kF,T) =

√a + b

[𝜌(lnkF −m

)+

√(lnkF −m)2 + s2

](2.4)

where a, b, 𝜌, m, and s are parameters depending on T, and kF = K/F isthe forward-moneyness. Intuitively a controls the overall level of variance,m corresponds to a moneyness shift, 𝜌 is related to the correlation betweenstock prices and volatility and controls symmetry, s controls the smoothnessnear the money (kF = 1), and b controls the angle between small and largestrikes.

Figure 2.6 shows an example of the shape of the smile produced by theSVI model, which is plausible.

The SVI model is connected to stochastic volatility models (see Gatheral-Jacquier (2011)). Specifically, the authors show how Equation (2.4) is thelimit-case of the implied volatility smile produced by the Heston model asthe maturity goes to infinity.

To ensure the no-arbitrage condition Equation (2.1) we must have b(1 +|𝜌|) ≤ 4T. In his original 2004 talk, Gatheral claims that this condition is

also sufficient to ensure Equation (2.2), but a recent report by Roper (2010)suggests otherwise.

An attractive property of the SVI model is that it is relatively easyto satisfy Equation (2.3) since its parameters are time dependent. This isalso a drawback: as a surface, the SVI model has too many parameters.


0.0 0.5 1.0 1.5 2.0

0.1

0.2

0.3

0.4

0.5

0.6

Forward moneyness kF

Implie

d v

ola

tilit

y σ

*

FIGURE 2.6 SVI fit of one-year implied volatility smile for the S&P 500 as ofJuly 18, 2012. Black dots correspond to observed data.

To circumvent this issue, Gurrieri (2011) put forward a class of arbitrage-freeSVI models with term structure using 11 time-homogenous parameters.

It should be noted that, being a function of ln KFand thus ln K

S, the SVI

model incorporates the sticky-delta rule and thus produces a higher deltathan Black-Scholes, as shown in Figure 2.7.

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

Delta o

f A

TM

call

Spot price (%Initial)

FIGURE 2.7 The SVI model of the implied volatility surface produces a higherdelta than Black-Scholes.


FOCUS ON SVI MODEL FITTING

The classical approach to find model parameters is to perform a leastsquare optimization against observed data. In the case of the SVImodel, this means finding:

mina,b,𝜌,m,s

n∑i=1

[𝜎∗2SVI

(ki,T; a,b, 𝜌,m, s

)− 𝜎∗2

Market(ki,T)

]2where T is a fixed listed maturity, k1,… , kn are n listed strikes in per-centage of the forward price, and 𝜎∗

Marketis the implied volatility of the

listed options observed on the market.When solving numerically, it is often necessary to specify bound-

aries on the parameters: a ≥ 0,b ≥ 0,−1 ≤ 𝜌 ≤ 1, s > 0. Additionallywe may include the no-arbitrage constraint b(1 + |𝜌|)T ≤ 4. Note thatwhile we could in principle let a < 0, this tends to generate bad results.

This will be enough for most optimization softwares. For fastercomputations, Zeliade Systems (2009) explains how the problem canbe reduced to a two-dimensional optimization problem with nestedlinear program, for which there is a quasi-explicit solution.

2-4.2 Indirect Models of Implied Volatility

Any alternative to Black-Scholes will generate an implied volatility surface,which may be used to appraise the quality of the model. This is a majorsource of implied volatility surface models.

2-4.2.1 The SABR Model The stochastic alpha, beta, rho (SABR) model ofHagan and colleagues (2002) assumes that the underlying forward pricedynamics are described by the coupled diffusion equations:{

dFt = 𝜎tF𝛽t dWt

d𝜎t = 𝛼𝜎tdZt

whereW and Z are standard Brownian motions with (dWt)(dZt) ≡ 𝜌dt and𝛼, 𝛽, 𝜌 are constant model parameters.


In the special case 𝛽 = 1 an analytical formula for implied volatility isavailable for short maturities, which allows to fit the parameters to observedoption prices. Near the money the formula has the Taylor expansion expres-sion:

𝜎∗SABR(k,T) ≈T→0k→1

𝜎0

[1 + 1

2𝜌𝛼 lnk𝜎0

+ 2 − 3𝜌2

12

(𝛼 lnk𝜎0

)2

+(14𝜌𝜎0𝛼 +

2 − 3𝜌2

12𝛼2

2

)T]

The SABR model is popular for interest rates where the smile is moresymmetric than for equities.

2-4.2.2 The Heston Model The Heston (1993) model is perhaps the mostpopular approach for stochastic volatility. It assumes the following underly-ing spot price dynamics coupled to an instantaneous variance process:{

dSt = 𝜇tStdt +√vtStdWt

dvt = 𝜅(v − vt

)dt + 𝜔

√vtdZt

whereW and Z are standard Brownian motions with (dWt)(dZt) ≡ 𝜌dt and𝜅, v, 𝜔, 𝜌 are constant model parameters that must satisfy the Feller condition2𝜅v ≥ 𝜔2 to ensure strictly positive instantaneous variance vt at all times.The instantaneous rate of return 𝜇t on the underlying asset can be anythingsince it will disappear under the risk-neutral measure.

Rouah (2013) provides a valuable resource detailing the theoretical andpractical aspects of the Heston model, including code examples.

Although no analytical formula for implied volatility is available,the popularity of the Heston model is largely due to the existence ofquasi-analytical formulas for European options making the computation ofimplied volatilities very quick.

Figure 2.8 compares a Heston fit to the S&P 500 implied volatility sur-face. We can see that the Heston model produces a plausible shape but is tooflat for short expiries.

One limitation of the Heston model is that instantaneous variance is asomewhat elusive concept, which cannot be measured in practice. Howeveran analytical formula for the total expected variance over a period [0, T] isavailable:

𝔼

(∫

T

0vtdt

)= vT + 1

𝜅(1 − e−𝜅T)(v0 − v) ≈ v0T


0.5

Time to expiry0.5

1

1.5

0.25

–0.25

–0.5

0

Log-strike k

FIGURE 2.8 Comparison of the S&P 500 implied volatility surface (top) with itsHeston fit (bottom) as of September 15, 2005.Source: Jim Gatheral, The Implied Volatility Surface: A Practitioner’s Guide(Hoboken, NJ: John Wiley & Sons, 2005). Reprinted with permission of JohnWiley & Sons, Inc.

Hence the expected total annualized variance is approximately constantat v0 regardless of maturity, which is inconsistent with empirical marketobservations (see Section 5-2).

FOCUS ON THE FELLER CONDITION

Diffusion processes of the form dXt = atdt + btdWt may a priori spanthe entire real line (−∞, +∞). However, we know that the geomet-ric Brownian motion obtained with at = 𝜇Xt,bt = 𝜎Xt remains strictlypositive ifX0 > 0.Whenmodeling quantities such as stock prices, inter-est rates, or instantaneous variance, it is often useful to ensure thatthese quantities remain bounded between certain levels 𝓁 and u (pos-sibly infinite). This will be the case when the coefficients at, bt satisfycertain conditions.

First, observe that if at, bt depend on Xt and take the value 0 at 𝓁and u the process is automatically bounded by [𝓁, u] by continuity ofdiffusion paths. This is a useful sufficient condition for boundednessbut it is by no means necessary.


The lower boundary 𝓁 is then classified as:

■ Attracting if, starting from x above 𝓁, the process (Xt) may hit𝓁 before an arbitrary level b > x with positive probability

■ Non-attracting otherwise

The same classification applies to the upper boundary u througha symmetric definition. Note that the hitting time could be infinite,which calls for a further distinction between attainable and unattain-able attracting boundaries. We refer the interested reader to Karlin andTaylor (1981).

When the diffusion is time-homogeneous with drift and volatil-ity coefficients a(x),b(x); that is, (Xt) is of the form dXt = a(Xt)dt +b(Xt)dWt, a characterization for the lower boundary 𝓁 to be non-attracting is:

limx↓𝓁 ∫

x0

xs(y)dy = ∞ s(y) = exp

(−∫

y

y0

2a (x)b2(x)

dx)

where x0 and y0 are arbitrary fixed points inside (𝓁, u).We now derive the corresponding Feller condition at 𝓁 = 0 for the

process dXt = 𝜅(𝜃 −Xt)dt + 𝜔√XtdWt so that the values remain in (0,

∞). Substituting the drift and volatility coefficients in the definition ofs(y) we have:

s(y) = exp(−∫

y

y0

2𝜅 (𝜃 − x)𝜔2x

dx)

=(y0y

)2 𝜅𝜃𝜔2

e2𝜅𝜔2

(y−y0)

Since limy→0

e2𝜅𝜔2

(y−y0) = e− 2𝜅𝜔2

y0 > 0 the improper integral ∫ x00 s(y)dy

will diverge if and only if 2 𝜅𝜃𝜔2

≥ 1, which is the Feller condition for theHeston model.

2-4.2.3 The LNV Model Recently Carr and Wu (2011) proposed a sophisti-cated framework for the underlying forward price dynamics, which resultsin a closed-form formula for implied volatility. The forward price processsolves the diffusion equation:

dFt =√vtFtdWt


where W is a standard Brownian motion and v is an arbitrary stochasticinstantaneous variance process. Furthermore, the entire implied volatilitysurface 𝜎∗

t (K,T) is assumed to evolve through time according to the diffusionequation:

d𝜎∗t (K,T) = 𝜇t(K,T)dt + 𝜔t(K,T)dZt

where Z is a standard Brownian motion with (dWt)(dZt) ≡ 𝜌tdt and 𝜇, 𝜔, 𝜌may be stochastic.

Carr-Wu then show that in this setup the implied volatility surface atany time t is fully determined by a quadratic equation, which depends on 𝜇t,𝜔t, 𝜌t and the unspecified vt.

This very general framework can produce a wide range of impliedvolatility surfaces. Choosing 𝜇t = 𝜅(𝜃 − 𝜎∗2

t (K,T)) and 𝜔t = we−𝜂(T−t)𝜎∗2t

(K,T) yields the Log-Normal Variance (LNV) model in which 𝜎∗2LNV,0(k,T)

is solution to the quadratic equation:

w2

4e−2𝜂T𝜎∗4

LNV,0(k,T) + [1 + 𝜅T +w2e−2𝜂TT − 𝜌w√ve−𝜂TT]𝜎∗2

LNV,0(k,T)

− [v + 𝜅𝜃𝜏 + 2w𝜌√ve−𝜂T lnk +w2e−2𝜂T ln2k] = 0

where 𝜅,w, 𝜂, 𝜃, v, 𝜌 are constant parameters and k denotes moneynessK/S0.The LNV model thus combines two attractive features: a functional

parametric form for the implied volatility, and known dynamics for the evo-lution of the underlying spot price as well as the implied volatility surfaceitself.

REFERENCES AND BIBLIOGRAPHY

Carr, Peter, and Liuren Wu. 2011. “A New Simple Approach for ConstructingImplied Volatility Surfaces.” Working paper, New York University and BaruchCollege.

Derman, Emanuel. 2010. “Introduction to the Volatility Smile.” Lecture notes,Columbia University.

Gatheral, Jim. 2004. “A Parsimonious Arbitrage-Free Implied Volatility Parameteri-zation with Application to the Valuation of Volatility Derivatives.” Proceedingsof the Global Derivatives and Risk Management 2004 Madrid conference.

Gatheral, Jim. 2005. The Implied Volatility Surface: A Practitioner’s Guide. Hobo-ken, NJ: John Wiley & Sons.

Gatheral, Jim, and Antoine Jacquier. 2011. “Convergence of Heston to SVI.”Quan-titative Finance 11 (8): 1129–1132.

Gurrieri, Sébastien. 2011. “A Class of Term Structures for SVI Implied Volatility.”Working paper. Available at http://ssrn.com/abstract=1779463 or http://dx.doi.org/10.2139/ssrn.1779463.

http://ssrn.com/abstract=1779463

http://dx.doi.org/10.2139/ssrn.1779463


Hagan, Patrick S., Deep Kumar, Andrew L. Lesniewski, and Diana E. Woodward.2002. “Managing Smile Risk.” Wilmott Magazine (September): 84–108.

Heston, Stephen L. 1993. “A Closed-Form Solution for Options with StochasticVolatility with Applications to Bond and Currency Options.” Review of Finan-cial Studies 6 (2): 327–343.

Hodges, Hardy M. 1996. “Arbitrage Bounds on the Implied Volatility Strike andTerm Structures of European-Style Options.” Journal of Derivatives (Summer):23–35.

Homescu, Cristian. 2011. “Implied Volatility Surface: Construction Methodologiesand Characteristics.” Available at http://arxiv.org/abs/1107.1834v1.

Karlin, Samuel, and Howard M. Taylor. 1981. “Diffusion Processes.” In A SecondCourse in Stochastic Processes, 157–396. San Diego, CA: Academic Press.

Lee, Roger. 2004. “The Moment Formula for Implied Volatility at Extreme Strikes.”Mathematical Finance 14: 469–480.

Roper, Michael. 2010. “Arbitrage Free Implied Volatility Surfaces.” Working paper.Available at www.maths.usyd.edu.au/u/pubs/publist/preprints/2010/roper-9.pdf.

Rouah, Fabrice. 2013. The Heston Model in Matlab and C#. Hoboken, NJ: JohnWiley & Sons.

Zeliade Systems. 2009. “Quasi-Explicit Calibration of Gatheral’s SVI model.” Zeli-ade White Paper.

PROBLEMS

2.1 No Call or Put Spread Arbitrage Condition

Consider an underlying asset S with spot price S and forward price F. Let rdenote the continuous interest rate for maturity T,U(S,K,T, r), L(S,K,T, r)be the upper and lower bounds on the slope of the smile correspondingto the no call or put spread arbitrage condition (2.1). Given 𝜕cBS

𝜕K=

−e−rTN(d2),𝜕cBS𝜕𝜎

= Ke−rT√TN′(d2), show that that U − L =

√2𝜋 exp(d2

1)

F√T

where d1,2 =ln(F∕K)± 1

2 𝜎∗2T

𝜎∗√T

.

2.2 No Butterfly Spread Arbitrage Condition

Assume zero interest rates and dividends. Consider the Black-Scholes for-mula for the European call struck at K with maturity T:

C(S,K,T, 𝜎) = SN(d1) − KN(d2) d1,2 =ln(S∕K) ± 1

2𝜎2T

𝜎√T

http://arxiv.org/abs/1107.1834v1

http://www.maths.usyd.edu.au/u/pubs/publist/preprints/2010/roper-9.pdf


where S is the underlying asset’s spot price, 𝜎 is the volatility parameter, andN(.) is the cumulative distribution function of a standard normal.

(a) Given 𝜕C𝜕K

= −N(d2),𝜕C𝜕𝜎

= K√TN′(d2), derive the identities:

■𝜕2C𝜕K2

=N′(d2)

K𝜎√T

■𝜕2C𝜕𝜎𝜕K

=d1𝜎N′(d2)

■𝜕2C𝜕𝜎2

=d1d2𝜎

K√TN′(d2)

(b) A second-order chain rule. Show that if f(x, y) and u(t) are C2 (twicecontinuously differentiable) then the second-order derivative of 𝜑(t) =f(t, u(t)) is given as:

𝜑′′ = fxx + 2 u′fxy + u′2fyy + u′′fy, i.e.:

𝜑′′(t) = fxx[t,u(t)] + 2u′(t) fxy[t,u(t)] + (u′(t))2fyy[t,u(t)] + u′′(t) fy[t,u(t)]

where fxx, fxy, and fyy denote the second-order partial derivatives of f.(c) Assume that the implied volatility smile 𝜎*(K) is C2 for a given matu-

rity T. Using your results in (a) and (b), show that for c(S,K,T) =C(S,K,T, 𝜎∗(K)):

𝜕2c𝜕K2

=N′(d2)

K𝜎∗(K)√T

[1 + 2d1

(K𝜎∗′ (K)

√T)+ d1d2

(K𝜎∗′ (K)

√T)2

+(K𝜎∗′′ (K)

√T)(K𝜎∗(K)

√T)

](d) What does the no butterfly arbitrage condition (2.2) reduce to?

2.3 Sticky True Delta Rule

Consider a one-year vanilla call with strikeK = 1, and let 𝜎∗(S) be its impliedvolatility at various spot price assumptions S. Assume zero rates and divi-dends and denote the call price:

c(S) = cBS(S, 𝜎∗(S)) = SN

(ln S + 1

2𝜎∗2 (S)𝜎∗(S)

)−N

(ln S − 1

2𝜎∗2 (S)𝜎∗(S)

)


(a) Show that the option’s delta Δ is

Δ(S) = dcdS

= N

(ln S + 1

2𝜎∗2 (S)𝜎∗(S)

)+ 𝜎∗′(S)N′

(ln S − 1

2𝜎∗2 (S)𝜎∗(S)

)

(b) Assume that 𝜎∗ is a linear function of Δ: 𝜎∗(S) = a + bΔ(S). Show that Δis solution to the first-order differential equation:

Δ = N⎛⎜⎜⎝ln S + 1

2

(a + bΔ

)2a + bΔ

⎞⎟⎟⎠ + bN′⎛⎜⎜⎝ln S − 1

2

(a + bΔ

)2a + bΔ

⎞⎟⎟⎠Δ′

(c) Is Δ higher or lower than the Black-Scholes delta?

2.4 SVI Fit

Using your favorite optimization software (Matlab, Mathematica, etc.) findthe parameters for the SVI model corresponding to a least square fit of thefollowing one-year implied volatility data:

Strike (%forward) 20% 50% 70% 90% 100% 110% 130% 150% 160%Implied volatility 45.5% 34.6% 29.4% 24.0% 22.3% 19.9% 16.4% 14.9% 14.3%

Answer: a = 0.0180, b = 0.0516, 𝜌 = −0.9443, m = 0.2960, s = 0.1350using initial condition a = 0.04, b = 0.4, 𝜌 = −0.4, m = 0.05, s = 0.1, andbounds a > 0, 0 < b < 2, −1 < 𝜌 < 1, s > 0.

CHAPTER 3Implied Distributions

Perhaps the favorite activity of quantitative analysts is to decode marketdata into information about the future upon which a trader can base hisor her decisions. This is the purpose of the implied distribution that trans-lates option prices into probabilities for the underlying stock or stock indexto reach certain levels in the future. In this chapter, we derive the implieddistribution and show how it may be exploited to price and hedge certainexotic payoffs.

3-1 BUTTERFLY SPREADS ANDTHE IMPLIED DISTRIBUTION

Vanilla option prices contain probability information about the market’sguess at the future level of the underlying asset S. For example, supposethat Kroger Co. trades at $24 and that one-year calls struck at $24 and $25trade at $1 and $0.60 respectively. If interest rates are zero, we may theninfer that the probability of the terminal spot price ST in one year to beabove $24 must satisfy:

ℙ{ST > 24} = 𝔼(I{ST>24})

≥1 − 0.6 = 0.4,

because the digital payoff I{ST>24} dominates the call spread as shown inFigure 3.1.

Similarly the price of a butterfly spread with strikes $23, $24, and $25(i.e., long one call struck at $23, short two calls struck at $24, and long onecall struck at $25) will give a lower bound for ℙ{23 < ST < 25} as shownin Figure 3.2.

33


Payoff

Final spot price ST

$24 $25

Call spread

Digital

FIGURE 3.1 The digital payoff dominates the call spread payoff.

Payoff

$23 $24

Butterfly

Digital spread

$25

Final spot price ST

FIGURE 3.2 The digital spread payoff dominates the butterfly spread payoff.

Generally, if all option prices are available along a continuum ofstrikes—such as option prices generated by an implied volatility surfacemodel—we may consider butterfly spreads with strikes K − 𝜀,K,K + 𝜀leveraged by 1/𝜀2 and obtain in the limit as 𝜀 → 0 the implied distributiondensity of ST:

ℙ{ST = K} = erT lim𝜀→0

c(K − 𝜀) − 2c(K) + c(K + 𝜀)𝜀2

= erTd2cdK2

(3.1)

where r is the continuous interest rate for maturity T, and c(K) denotes theprice of the call struck at K.

Note that by put-call parity we have d2cdK2 = d2p

dK2 and thus put prices mayalternatively be used to compute the implied distribution density.

Figure 3.3 compares the Black-Scholes lognormal density to the implieddistribution density generated by an SVI model fit of S&P 500 option priceswith about 2.4-year maturity. We can see that the implied density is skewedtoward the right and has a fatter left tail.

Implied Distributions 35

Density

Strike (moneyness)

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Lognormal Implied

FIGURE 3.3 The lognormal and implied distribution densities.

The implied distribution density may be obtained directly from a smoothvolatility surface 𝜎∗(K,T) by differentiating c(K) = cBS(S,K,T, r, 𝜎∗(K,T))twice with respect to K (see Problem 2.2 (c)). The corresponding formula is:

ℙ{ST = K} =N′(d2)

K𝜎∗√T

[1 + 2d1

(K𝜕𝜎∗

𝜕K

√T)+ d1d2

(K𝜕𝜎∗

𝜕K

√T)2

+(K𝜕2𝜎∗

𝜕K2

√T)(K𝜎∗

√T)

](3.2)

where d1,2 =ln(F∕K)± 1

2 𝜎∗2T

𝜎∗√T

, F is the forward price of S for maturity T, and

N′(⋅) is the standard normal distribution density. It is worth emphasizing thatEquation (3.2) relies on partial derivatives of 𝜎* with respect to the dollarstrikeK, and that one must be careful when using an implied volatility modelsuch as the SVI model, which is based on forward-moneyness kF = K/F. In

the latter case, we must substitute 𝜕𝜎∗

𝜕K= 1

F×𝜕𝜎∗

SVI𝜕kF

and 𝜕2𝜎∗

𝜕K2 = 1F2

×𝜕2𝜎∗

SVI

𝜕k2F.

The first factor, N′(d2)K𝜎∗

√T, is the Black-Scholes lognormal distribution at

pointK using implied volatility.Without the second factor between brackets,the integral does not sum to 1, unless the smile is flat.

The implied distribution reveals what options markets “think” in termsof the future evolution of the underlying asset price. It is a useful theoretical


concept, but in practice it can be difficult to exploit this informationfor trading.

3-2 EUROPEAN PAYOFF PRICING AND REPLICATION

Consider an option with arbitrary European payoff f(ST) at maturity T, andlet h(K) = ℙ{ST = K} be the implied distribution density. The corresponding

option value is then f0 = e−rT𝔼(f (ST)) = e−rT∫∞

0f (K)h(K)dK and by direct

substitution of Equation (3.1) we obtain the Breeden-Litzenberger (1978)formula:

f0 = ∫∞

0f (K) d

2cdK2

dK

The knowledge of the implied distribution thus allows us to value anyEuropean option consistently with the vanilla option market. It turns outthat this value is in fact an arbitrage price, at least in theory when we cantrade all vanilla options along a continuum of strikes K > 0.

To see this, assume that f is bounded and smooth (twice continuouslydifferentiable) to perform an integration by parts and write:

f0 = ∫∞

0f (K)c′′(K)dK = [f (K)c′(K)]∞0 − ∫

∞

0f ′(K)c′(K)dK

= −f (0)c′(0) − ∫∞

0f ′(K)c′(K)dK

because infinite-strike calls are worthless. Integrating by parts again yields:

f0 = −f (0)c′(0) + f ′(0)c(0) + ∫∞

0f ′′(K)c(K)dK

Furthermore zero-strike calls are always worth c(0) = e−rTF; additionally,by put-call parity c(K) = p(K) + e−rT(F − K), thus by differentiation c′(K) =p′(K) − e−rT and since zero-strike puts are worthless we have c′(0) = −e−rT .Substituting into the previous equation we get:

f0 = f (0)e−rT + f ′(0)Fe−rT + ∫∞

0f ′′(K)c(K)dK

This expression suggests that the option may be hedged with a portfolio:

■ Long zero-coupon bonds in quantity f(0)e–rT■ Long zero-strike calls in quantity f ′(0)■ Long all vanilla calls struck at K > 0 in quantities f ′′(K)dK


The definite proof of this result is provided by establishing that theoption payoff perfectly matches the portfolio payoff:

f (ST) = f (0) + f ′(0)ST + ∫∞

0f ′′(K)max(0, ST − K)dK

which simply turns out to be the first-order Taylor expansion of f over [0,ST] with remainder in integral form after noticing that the bounds of theintegral are actually 0 and ST.

An alternative approach mixing puts and calls is developed inProblem 3.2.

This is a strong fundamental result that directs how to price and hedgeEuropean payoffs, with the following limitations:

■ It only applies to European payoffs. Other option payoffs that dependon the history of the underlying asset price (such as Asian or barrieroptions) or have an early exercise feature (such as American options)require more sophisticated valuation methods and cannot be perfectlyreplicated with a static portfolio of vanilla options.

■ In practice, only a finite number of strikes are available. While it is possi-ble to overhedge convex payoffs with a finite portfolio of vanillas, exactreplication cannot be achieved.

FOCUS ON OVERHEDGING

Exotic option traders often look at ways to overhedge a particularoption payoff with a portfolio of vanillas, and price it accordingly.They will then underwrite the exotic option and buy the vanilla port-folio on the option market; at maturity, any difference in value will bea positive profit.

Consider for example the exotic option payoff f (ST) = min(1,

S2T

S20

)(“capped quadratic”). Using the implied distribution shown inFigure 3.3 we find a theoretical price of 79%.

The following are two possible overhedging strategies:

■ An at-the-money covered call (i.e., long one stock and short anat-the-money call). The cost of this strategy is 1 – c(1) ≈ 87%,which is significantly more expensive than the theoretical price.

■ Long 1/2 stock, long one call struck at 50%, short 1.5 at-the-money calls. The cost of this strategy is 1/2 + c(0.5) – 1.5 ×c(1) ≈ 81%, which is much closer to the theoretical price.


Figure 3.4 compares the payoffs of the exotic option versus thetwo overhedging portfolios.

Following the methodology of Demeterfi, Derman, Kamal, andZhou (1999), in general any convex portion of a payoff f(ST) over

0.5 1.0 1.5 2.0

1.0

0.8

0.6

0.4

0.2

FIGURE 3.4 Payoffs of an exotic option and two possible vanilla overhedges.

an interval [S– , S+] may be overhedged using, for example, n vanillacalls struck at S− = K1 < K2 < · · · < Kn < S+ in quantities q1,… , qnand zero-coupon bonds in quantity e−rTf (S−) such that:⎧⎪⎪⎨⎪⎪⎩

f(S−

)+

n−1∑i=1

qimax(0,Kj − Ki) = f (Kj) for all 2 ≤ j ≤ n

f (S−) +n∑i=1

qimax(0, S+ − Ki) = f (S+)

We may then bootstrap the quantities qi as follows:

q1 =f (K2) − f (K1)K2 − K1

q2 =f (K3) − f (K2)K3 − K2

− q1

⋮

qn =f (S+) − f (Kn)S+ − Kn

−∑n−1

i=1qi


In our capped quadratic option example over [0, 1] with strikesspaced 0.25 apart, we find quantities q1 = 0.25 and q2 = q3 = q4 =0.5, and the convex portion of the portfolio payoff is very close to theactual payoff as shown in Figure 3.5.

0.2 0.60.4 0.8 1.0

1.0

0.8

0.6

0.4

0.2

FIGURE 3.5 Overhedging the “capped quadratic” option with calls.

3-3 PRICING METHODS FOR EUROPEAN PAYOFFS

As stated earlier, the implied distribution h(K) = ℙ{ST = K} makes it possi-ble to price any European payoff f(ST). Several numerical integrationmethods, such as the trapezoidal method, are then available to compute

f0 = e−rT∫∞

0f (K)h(K)dK. One issue is that these methods become ineffi-

cient in large dimensions (i.e., multi-asset payoffs), which is the topic ofChapters 6 to 9.

Another approach isMonte Carlo simulation, which is easy to generalizeto multiple dimensions. Using a cutoff A≫ 0 we may approximate f0 by:

f0 ≈ e−rT∫A

0f (u)h(u)du ≈ e−rT

nA

n∑i=1

f (Aui)h(Aui)

where u1,… , un are n independent simulations from a uniform distributionover [0, 1].

We can improve this approach by means of the importance sam-pling technique, which exploits the fact that the implied distribution is


bell-shaped and somewhat similar to the Black-Scholes lognormal distribu-tion 𝓁(K) = 1

K𝜎√TN′(d2) with sensible volatility parameter 𝜎 (e.g., at-the-

money implied volatility for maturity T). To do so, rewrite:

f0 = e−rT∫∞

0f (u)h(u)

𝓁(u)𝓁(u)du = e−rT𝔼

[f (X) h(X)

𝓁(X)

]≈ e−rT

n

n∑i=1

f (xi)h(xi)𝓁(xi)

whereX is lognormally distributed with density 𝓁(K) and x1,… , xn are sim-ulated values ofX. The ratio h(X)∕𝓁(X)measures how close the implied andlognormal distributions are.

Simulating X is straightforward through the identity X = F ×exp

(𝜀𝜎

√T − 1

2𝜎2T

)where 𝜀 is a standard normal, for which there are

many efficient pseudo-random generators. For completeness we providethe Matlab code to price the “capped quadratic” payoff according to theimplied distribution shown in Figure 3.3.

function price = ImpDistMC(n)%Note: this algorithm assumes zero rates and dividends

epsilon = 0.0001;

T = 2.41; %maturity

function val = Payoff(S)val = min(1, S∧2);end

function vol = SVI(k)a = 0.02;b = 0.05;rho = -1;m = 0.3;s = 0.1;

k = log(k);vol = sqrt( a + b*( rho*(k-m) + sqrt( (k-m)∧2 + s∧2 ) ) );

end

function density = ImpDist(k)vol = SVI(k);volPrime = (SVI(k+epsilon)-vol)/epsilon;

%finite difference

volDblPrime = (SVI(k+epsilon)-2*vol+SVI(k-epsilon))/epsilon∧2;%finite diff.

d1 = (-log(k)+0.5*vol∧2*T)/(vol*sqrt(T));


d2 = d1 - vol*sqrt(T);

density = normpdf(d2)/(k*vol*sqrt(T)) * (1 + ...2*d1*k*volPrime*sqrt(T) + ...d1*d2*(k*volPrime*sqrt(T))∧2 + ...(k*volDblPrime*sqrt(T))*(k*vol*sqrt(T)) );

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MAIN FUNCTION BODY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

atm_vol = SVI(1);

price = 0;for i=1:nx = exp(atm_vol*randn*sqrt(T) - 0.5*atm_vol∧2*T);

%lognormal simulation

price = (i-1)/i*price + Payoff(x)*ImpDist(x) .../ lognpdf(x,-0.5*atm_vol∧2*T,atm_vol*sqrt(T)) / i;

endend

3-4 GREEKS

Because the implied distribution is derived from vanilla option prices in thefirst place, it produces the same delta as the input volatility smile. Conse-quently, if the implied volatility smile incorporates a sticky-delta rule, thedelta obtained by repricing calls using the implied distribution will be higherthan the Black-Scholes delta.

The implied distribution also makes it possible to calculatesmile-consistent Greeks for any European payoff. In practice this isoften done using finite differences, but one must be careful with the numer-ical precision of the method used for pricing. In particular the Monte Carlomethod has error of order 1∕

√n where n is the number of simulations,

and will generate a different price on each call unless the pseudo-randomgenerator is systematically initialized at the same seed.

It is worth emphasizing that the implied distribution depends on thevariables or parameters used for the Greeks. For example, if the payoff f(ST)is independent from the initial spot price S0 the delta may be written 𝜕f0

𝜕S0=

e−rT∫∞

0f (x) 𝜕

𝜕S0ℙ{ST = x}dx which involves the derivative of the implied

distribution with respect to the spot price.


REFERENCES

Breeden, Douglas T., and Robert H. Litzenberger. 1978. “Prices of State-ContingentClaims Implicit in Option Prices.” Journal of Business 51 (4): 621–651.

Demeterfi, Kresimir, Emanuel Derman, Michael Kamal, and Joseph Zhou. 1999.“More than You EverWanted to Know about Volatility Swaps.”Goldman SachsQuantitative Strategies Research Notes, March.

PROBLEMS

3.1 Overhedging Concave Payoffs

Consider a concave payoff f(ST) over an interval [S– , S+]. Propose a methodto overhedge this payoff using a finite portfolio of vanilla calls.

3.2 Perfect Hedging with Puts and Calls

Show that for any smooth (twice continuously differentiable) payoff f(ST):

f (ST) = 𝛼 + 𝛽ST + ∫F

0f ′′(K)max(0,K − ST)dK

+ ∫∞

Ff ′′(K)max(0, ST − K)dK

where F is the forward price and 𝛼 and 𝛽 are constants to be identified. Whatis the corresponding option price f0?

3.3 Implied Distribution and Exotic Pricing

(a) Reproduce Figure 3.3 using T = 2.4 and the following parameters forthe SVI model: a = 0.02, b = 0.05, 𝜌 = –1, m = 0.3, s = 0.1. Assume S0= 1 as well as zero interest and dividend rates.

(b) Using a numerical integration algorithm check the price of the “cappedquadratic” option, then compute the price of the following optionpayoffs:

■ f (ST) = max(0,ST − 1ST

)


■ f (ST) =

{max

(1 + C,

√ST

)if ST > 75%

ST otherwise, where C = 25%. Then

solve for C to obtain a price of 100%.■ f (ST) = − 2

Tln ST . What is

√f0?

■ f (ST) = max(0, (ST − 1)3). Then find a vanilla overhedge over the inter-val [0,2] with strikes 0.5, 0.9, 1, 1.1, 1.3, 1.7 and compare the price ofthe overhedge with the theoretical price.

3.4 Conditional Pricing

On August 24, 2012, Kroger Co.’s stock traded at $21.795, and optionsmaturing in 511 days had the following implied volatilities:

Strike (%Spot) 30% 50% 70% 90% 100% 110% 130% 150% 200%

Impl. Vol. (%) 49.58 36.59 30.17 25.43 24.23 22.97 21.40 20.86 22.89

The forward price was $21.366, and the continuous interest rate was0.81%.

(a) Calibrate the SVI model parameters to this data. Answer: a = 0, b =0.1272, 𝜌 = – 0.7249, m = –0.1569, s = 0.5388 using initial conditiona = 0.04, b = 0.4, 𝜌 = –0.4, m = 0.05, s = 0.1, and bounds a > 0, 0 < b< 2.024, –1 < 𝜌 < 1, s > 0.

(b) Produce the graph of the corresponding implied distribution and com-pute the price of the “capped quadratic” option with payoff f (ST) =min(1, (ST∕S0)2) using the numerical method of your choice. Answer:approximately $0.797.

(c) The following graph (Figure 3.6) shows the history of Kroger Co.’s stockprice since 1980. Based on this graph, you reckon that the stock pricewill remain above $14 within the next three years.i. Compute the probability that ST > 14 using the implied distribution.ii. Compute the value of the “capped quadratic” option conditional

upon {ST > 14}. Is this an arbitrage price?


0

5

10

15

20

25

30

35

40

Oct-8

0

Oct-8

2

Oct-8

4

Oct-8

6

Oct-8

8

Oct-9

0

Oct-9

2

Oct-9

4

Oct-9

6

Oct-9

8

Oct-0

0

Oct-0

2

Oct-0

4

Oct-0

6

Oct-0

8

Oct-1

0

FIGURE 3.6 Historical price of Kroger Co.’s stock since 1980.

3.5 Path-Dependent Payoff

Consider an option whose payoff at maturity T2 is a nonlinear functionf (ST1, ST2

) of the future underlying spot price observed at times T1 < T2.

(a) Give two classical examples of such an option.(b) Write the pseudo-code to price this option in the Black-Scholes model

using the Monte Carlo method.(c) Assume that the implied distributions of both ST1

and ST2are known.

Can you think of a method to find “the” value of the option? If yes pro-vide the pseudo-code; if not explain what information you are missing.

3.6 Delta

Modify the code from Section 3-3 to calculate the delta of the “cappedquadratic” option. Hint: Write 𝜎∗(S0,K) ≡ 𝜎∗(K∕S0) and carefully amendEquation (3.2) accordingly.

CHAPTER 4Local Volatility and Beyond

The local volatility model was independently developed in the early 1990sby Derman and Kani and by Dupire. It has arguably become the benchmarkmodel to price and hedge a wide range of equity exotics such as digitals,Asians, and barriers, but fails on certain payoffs such as forward startoptions, which are better approached using a stochastic volatility model.The model can be difficult to implement since it requires a high-quality,smooth implied volatility surface as input, and simulation of all intermediatespot prices until maturity using short time steps.

4-1 LOCAL VOLATILITY TREES

The local volatility model is best visualized on a binomial tree: instead ofusing the same volatility parameter to generate the tree of future spot prices(Figure 4.1) the local volatility model uses a different volatility parameter atevery node (Figure 4.2). The option is then priced as usual using backwardinduction.

Given a local volatility function 𝜎loc(t,S) it is relatively easy to construct atree and then compute the correspondingmodel-implied volatilities 𝜎*(K,T).A step-by-step guide can be found in Derman and Kani (1994).

In practice we are faced with the reverse problem: given market-impliedvolatilities 𝜎*(K, T) for a finite set of strikes and maturities, we want to findthe corresponding local volatilities 𝜎loc(t, S). This leads to an unstable cali-bration problem where a small change in input may produce very differentresults.

We will not venture into these topics, mostly because finite differenceand Monte Carlo methods are now preferred to tree methods.

45


FIGURE 4.1 Binomial tree with constant volatility.

FIGURE 4.2 Binomial tree with local volatility.

4-2 LOCAL VOLATILITY IN CONTINUOUS TIME

The continuous time formulation of the local volatility model is that the spotprice process is a diffusion of the form:

dSt∕St = 𝜇tdt + 𝜎loc(t, St)dWt

where the drift μt is irrelevant because it will disappear under the risk-neutralmeasure, and 𝜎loc(t, S) is a function which is uniquely determined when theentire implied volatility surface 𝜎*(K, T) is known.

In practice it is best to start with a smooth implied volatility surface suchas an SVI fit to option market data and compute local volatilities by meansof Dupire’s equation (see Section 4-3.1).

Local Volatility and Beyond 47

With the local volatility function in hand, time-dependent exotic pay-offs may be priced using Monte Carlo simulations. The Euler-Maruyamadiscretization method provides an easy way to simulate a single path alongdiscrete time steps of length Δt by iterating the formula:

St+Δt = St[1 + 𝜈tΔt + 𝜎loc

(t, St

)�̃�t√Δt

]where 𝜈t is the risk-neutral drift (forward interest rate minus dividend rate)at time t and �̃�t is a standard normal.

Figure 4.3 compares a simulated path using the local volatility modelwith its corresponding geometric Brownian motion. We can see that the twopaths differ somewhat significantly.

It is worth noting that the Euler-Maruyama method might occasionallyproduce negative prices, particularly when Δt is too large. This will happenwhenever �̃�t <

−1−𝜈tΔt𝜎loc(t,St)

√Δt.

More efficient discretization methods such as Milstein’s are also avail-able in order to improve the accuracy of simulated paths, at the cost ofadditional computations.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Spo

t pric

e

Time

Simulationwith localvolatility

GeometricBrownianmotion

FIGURE 4.3 Simulated path with local volatility and corresponding geometricBrownian motion.


4-3 CALCULATING LOCAL VOLATILITIES

4-3.1 Dupire’s Equation

When all call prices c(K, T) are available for a continuum of strikes K > 0and maturities T > 0—such as call prices generated by an implied volatilitysurface model—the local volatilities are given by Dupire’s equation:

𝜎2loc(T,K) =

𝜕c𝜕T

12K2 𝜕2c𝜕K2

(4.1)

provided interest and dividend rates are zero and c is C2,1 (twice continu-ously differentiable with respect to K and continuously differentiable withrespect to T). In other words, local volatility at time T and level K is pro-portional to the price ratio of an instant calendar spread to an infinitesimalbutterfly spread.

We can get the intuition behind Equation (4.1) by rewriting c(K,T) =c(t, St,K,T) and making the approximation that, conditional upon ST = K,the local volatility 𝜎loc(T, K) is constant over the interval [T,T + ΔT]. Then:

■ Using the rule of thumb that for constant volatility 𝜎 an at-the-moneycall is approximately worth1 c(t,K,K,T) ≈ 1√

2𝜋K𝜎

√T − t, we get

𝜕c𝜕T

(t,K,K,T) ≈ K

2√2𝜋(T−t)𝜎 and thus 𝜕c

𝜕T(T,K,K,T + ΔT) ≈ K

2√2𝜋ΔT

𝜎loc(T,K).■ Ignoring time value of money, the denominator is proportional tothe implied distribution 𝜕

2c𝜕K2 , which reduces to the Black-Scholes log-

normal distribution N′(d2)K𝜎

√T−t

when volatility is constant. Thus 12K2

𝜕2c𝜕K2 (T,K,K,T+ΔT)≈ 1

2K2 N′(d2)

K𝜎loc(T,K)√ΔT

≈ K

2√2𝜋ΔT

1𝜎loc(T,K)

since N′(d2)≈1√2𝜋

at the money.

Taking the ratio of the two quantities and simplifying, we get 𝜎2loc(T,K)

as required. A more rigorous derivation of Equation (4.1) is given inAppendix 4.A.

In practice, local volatilities can be difficult to compute, particularly fordeeply out-of-the-money strikes and when input call prices are directly takenfrom the market without careful interpolation techniques. In particular,

1See, for example, Bossu and Henrotte (2012), Chapter 7, Problem 8.


approximating 𝜕2c𝜕K2 with finite differences can create instability because this

quantity is very small for deeply out-of-the-money strikes.There are two standard approaches to address this issue:

1. Specify local volatility in functional parametric form and calibrate tooption prices

2. Specify the implied volatility surface in functional parametric form andcompute the corresponding local volatility surface

We now describe the second approach.

4-3.2 From Implied Volatility to Local Volatility

Assuming that the implied volatility surface 𝜎∗(K,T) is known and smooth,we can directly compute the corresponding local volatility surface by substi-tuting into Equation (4.1) the derivatives of c(K,T) = cBS(S,K,T, 𝜎∗(K,T))(see Problem 4.1). After calculations we obtain:

𝜎2loc(T,K) = 𝜎∗2

×1 + 2T𝜎∗𝜕𝜎∗

𝜕T

1 + 2d1(K𝜕𝜎∗

𝜕K

√T)+ d1d2

(K𝜕𝜎∗

𝜕K

√T)2

+(K𝜕2𝜎∗

𝜕K2

√T)(K𝜎∗

√T)(4.2)

where for ease of notation we omitted evaluations at (K, T) for 𝜎* andits partial derivatives. Note that rates and dividends are again assumed tobe zero.

As unpalatable as it may look, this formula tends to be relatively sta-ble numerically, provided the implied volatility surface supplied as inputis smooth. Figure 4.4 compares the implied and local volatility surfacesobtained from Gurrieri’s (2010) model fitted to S&P 500 option prices. Wecan see that for fixed maturity the slope or skew of the local volatility curveis steeper.

4-3.3 Hedging with Local Volatility

The local volatility model produces the same delta as the implied volatil-ity surface, which is higher than the Black-Scholes delta using a sticky-moneyness or sticky-delta rule. This consistency is another attractive featureof the local volatility model.


Maturity

Moneyness(strike / spot)

Vol

.

2.0

1y

2y

1.5

1.0

0.5

0.00.0

0.51.0

1.52.0

FIGURE 4.4 Local volatility surface (top) vs. implied volatility surface (bottom),fitted to S&P 500 option prices as of August 29, 2012. Implied volatilities wereshifted down by 0.1 to better separate the two surfaces.

4-4 STOCHASTIC VOLATILITY

As stated in the introduction, the local volatility model can be used to price awide range of equity exotics, but not all. In particular, forward start and cli-quet options depend heavily on the future evolution of the implied volatilitysurface and require a more sophisticated approach.

There are many ways to define stochastic volatility, but most modelstend to specify joint spot price and instant volatility dynamics, that is:

■ A stochastic differential equation for the underlying spot price process,or sometimes the forward price process, which is typically of the form:

dSt∕St = 𝜇(· · ·)dt +√vtdWt (4.3)

where W is a standard Brownian motion, 𝜇 can be anything, and vtdenotes instant variance;

■ A stochastic differential equation for instant variance, typically of theform:

dvt = 𝛼tdt + 𝜔tdZt (4.4)

where Z is a standard Brownian motion with correlation (dWt)(dZt) ≡𝜌tdt, 𝛼t is the instant variance drift, and 𝜔t is the volatility of instantvariance. Note that in general 𝛼,𝜔, 𝜌 could be stochastic processes drivenby other sources of randomness than W and Z.


The Heston and Carr-Wu LNV models (see Section 2-4.2.3) areexamples of such models under risk-neutral probabilities.

To a degree, the local volatility model may be seen as an elementarystochastic volatility model in which instant variance vt = 𝜎2loc(t, St) hasperfect correlation of ±1 with the stock price. When correlation is notperfect, a connection persists in the form of conditional expectations(see Section 4-4.2). As such the local volatility function may be seen as thebackbone of every stochastic volatility model of the form (4.3) and (4.4).

Note that some authors focus on (4.3) without specifying (4.4) (e.g.,Carr and Wu (2011)) while others ditch (4.3) and only focus on (4.4)(e.g., Bossu (2005)).

Because instant variance is not a tradable asset (nor even an observablequantity), its drift may be of significance, and modelers often stipulate meanreversion in its definition. This is the case of the Heston model, whose drift𝜅(v − vt) will be more strongly positive or negative as vt goes further awayfrom the long-run variance v, pulling vt back towards v.

Note, however, that with the development of variance swap markets

(see Section 5-2), total varianceVT = ∫ T0 vtdt ≡ ∫ T

0

(ln St+dt

St

)2dt has become

tradable. We may thus compare 𝔼(VT) = ∫ T0 𝔼(vt)dt to market prices to cal-

ibrate the model and assess its quality.

4-4.1 Hedging Theory

When the instant variance process is an Ito process, one major theoreticaladjustment caused by stochastic volatility models of the form (4.3) and (4.4)is the addition of an ancillary option as hedging instrument.

We now provide a slightly modified version of the hedging argument fol-lowed by Wilmott (2006). Assuming that all option values f (t, St, vt) dependonly on time, spot, and instant volatility, we have by the multidimensionalIto-Doeblin theorem:

dft =𝜕f𝜕tdt +𝜕f𝜕SdSt +𝜕f𝜕vdvt +

12

[𝜕2f

𝜕S2(dSt

)2+ 2𝜕2f𝜕S𝜕v

(dSt)(dvt) +𝜕2f

𝜕v2(dvt)2

],

that is:

dft =[𝜕f𝜕t

+ 12𝜕2f

𝜕S2vtS

2t +

12𝜕2f

𝜕v2𝜔2t +𝜕2f𝜕S𝜕v𝜌t𝜔t

√vtSt

]dt +𝜕f𝜕SdSt +𝜕f𝜕vdvt.

Because dft is exposed to two sources of risk, namely dSt contingent onW and dvt contingent on Z, which is correlated with W, perfect hedgingrequires trading S and another option g on the same underlying asset S withvalue g(t, St, vt). Specifically, we must sell a quantity 𝛿g =

𝜕f𝜕v

/ 𝜕g𝜕v

of g to cancel


dvt terms; the variation in value Π of the corresponding portfolio long oneoption f and short 𝛿g options g is then:

dΠt = [𝛾f,t − 𝛿g 𝛾g,t]dt +(𝜕f𝜕S

− 𝛿g𝜕g𝜕S

)dSt

where 𝛾f,t =𝜕f𝜕t

+ 12𝜕2f𝜕S2

vtS2t +

12𝜕2f𝜕v2

𝜔2t +

𝜕2f𝜕S𝜕v

𝜌t𝜔t√vtSt is the drift coefficient

for option f and 𝛾g,t is defined similarly with respect to option g.

We may now sell a quantity 𝛿S =𝜕f𝜕S

− 𝛿g𝜕g𝜕S

of S to obtain a riskless port-folio long one option f, short 𝛿g options g and short 𝛿S units of S, whosevalue P must grow at the risk-free rate under penalty of arbitrage. Thus,

r(ft − 𝛿ggt − 𝛿SSt) = 𝛾f,t − 𝛿g 𝛾g,t

Substituting 𝛿S =𝜕f𝜕S

− 𝛿g𝜕g𝜕S, 𝛿g =

𝜕f𝜕v

/ 𝜕g𝜕v

and rearranging terms we obtainthe remarkable identity:

rft − rSt𝜕f𝜕S

− 𝛾f,t

𝜕f𝜕v

=rgt − rSt

𝜕g𝜕S

− 𝛾g,t

𝜕g𝜕v

Because the left-hand side only depends on f while the right-hand sideonly depends on g, and because f and g can be chosen arbitrarily, we concludethat all option values must satisfy the pricing equation:

rft − rSt𝜕f𝜕S

− 𝛾f,t = Λt𝜕f𝜕v

for some “universal” function Λt = Λ(t, St, vt) common to all options.It turns out that Λt is the risk-neutral drift of instant variance. Indeed,

under risk-neutral probabilities we must have 𝔼t(dSt) = rStdt and 𝔼t(dft) =rftdt. Taking conditional expectations of:

dft = 𝛾f,tdt +𝜕f𝜕SdSt +

𝜕f𝜕vdvt

and making these substitutions we get rftdt = 𝛾f,tdt +𝜕f𝜕SrStdt +

𝜕f𝜕v𝔼t(dvt),

leaving 𝔼t(dvt) = Λtdt after all cancellations.In theory Λt could be implied from option prices, but this is very difficult

to achieve in practice. Instead, most models specify the risk-neutral dynamicsof instant variance directly, which amounts to specifying Λt. In the case ofthe Heston model we have:

Λt = Λ(t, St, vt) = 𝜅(v − vt)


which is algebraically independent from the spot price St. Note, however,that St and vt are not statistically independent if 𝜌t ≢ 0.

4-4.2 Connection with Local Volatility

It can be shown (see, e.g., Gatheral (2005), who follows Derman and Kani(1997)) that within the framework (4.3) and (4.4) the conditional expecta-tion of stochastic instant variance vt upon the spot price St must be the localvariance:

𝔼(vt|St = S) = 𝜎2loc(t, S)

This is a remarkable property, which further underscores the significanceof the local volatility concept.

It should be emphasized that stochastic volatility models such as Hes-ton typically violate this property when local volatilities are computed fromoption market prices. This will happen with all models that do not exactlymatch the market’s implied volatility surface.

As an interesting corollary, by iterated expectations the undiscountedprice of total variance reduces to the expected total local variance:

𝔼

(∫

T

0vtdt

)= 𝔼

[∫

T

0𝜎2loc

(t, St

)dt

]which explains why the local volatility model correctly prices variance swaps(see Section 5-2) even though it does not provide a correct replication strat-egy. Note how the expectation on the right-hand side need not involve vat all.

4-4.3 Monte Carlo Method

An attractive feature of stochastic volatility models is that, once theyhave been satisfactorily calibrated to vanilla options prices, it is relativelyeasy to price exotics by Monte Carlo simulations. This may be done byEuler-Maruyama discretization of Equations (4.3) and (4.4) with short timesteps of length Δt:⎧⎪⎨⎪⎩

vt+Δt = vt + ΛtΔt + 𝜔t(𝜌t�̃�1,t +

√1 − 𝜌2t �̃�2,t

)√Δt

St+Δt = St[1 + (r − q) Δt +

√vt�̃�1,t

√Δt

]where Λt is the risk-neutral drift of vt, r and q are respectively the continuousinterest and dividend rates, and �̃�1,t, �̃�2,t are independent standard normals.

In practice the Euler-Maruyama scheme may produce negative instantvariance when Δt is too large or Feller conditions are disregarded.


A quick-and-dirty fix is then to reset vt at zero, but for some models thismight mean that vt gets stuck there, so this fix should be used with caution.Another common fix is to take absolute values.

More sophisticated simulation methods targeting the Heston modelhave been put forward in recent years. A good reference is Andersen (2007),which reviews several schemes and develops new ones.

4-4.4 Pricing and Hedging Forward Start Options

Recall that a forward start call option with maturity T has its strike set ata future date 0 < t0 < T, usually as a percentage k of the future spot priceSt0 . For example consider a one-year at-the-money call starting one monthforward. After a month, this option will become a regular one-year2 vanillacall with strike K = S1∕12, at which point it will be approximately worthS1∕12√

2𝜋𝜎∗(S1∕12,1) if interest and dividend rates are close to zero.

To price the forward start call we thus need to compute 1√2𝜋e−r∕12𝔼[S1∕12

𝜎∗(S1∕12,1)], whence the relevance of stochastic volatility models. Note thatthe use of the risk-free interest rate r, rather than a higher rate adjusted forrisk, is justified by the theoretical existence of a riskless hedging portfolio.

In practice it turns out that the price of an at-the-money forward startoption is not significantly different using a stochastic volatility model such asHeston or using the local volatility model. This is because the price is approx-imately linear in volatility. For out-of-the-money forward start options, how-ever, the difference is more significant because there is some convexity involatility known as volga.

In terms of hedging, it should be emphasized that stochastic volatilitymodels rely on both continuous delta- and vega-hedging using the underlyingasset and another option. This is very impractical because of higher trans-action costs that are incurred on option markets and exotic traders tend toonly execute the initial vega-hedge and charge extra for residual volga risk.

4-4.5 A Word on Stochastic Volatility Modelswith Jumps

A popular theoretical extension for stochastic volatility models is the addi-tion of jumps; that is, sudden spot price discontinuities. We do not discuss

2Note that in this example the actual maturity of the forward start option is13 months at inception. In some cases the nominal maturity might mean the actualmaturity, from which the forward start period should be subtracted, so one mustalways double-check to see which convention applies.


these models, mostly because jumps of random size cannot be hedged andthe riskless dynamic arbitrage argument thus breaks down. Two accessiblereferences here are Gatheral (2005) and Wilmott (2006).

REFERENCES

Andersen, Leif B. G. 2007. “Efficient Simulation of the Heston Stochastic Volatil-ity Model.” Working paper. Available at http://ssrn.com/abstract=946405 orhttp://dx.doi.org/10.2139/ssrn.946405.

Bossu, Sébastien. 2005. “Arbitrage pricing of equity correlation swaps.” JPMorganEquity Derivatives Report.

Bossu, Sébastien, and Philippe Henrotte. 2012. An Introduction to EquityDerivatives: Theory and Practice, 2nd ed. Chichester, UK: John Wiley &Sons.

Carr, Peter and Liuren Wu. 2011. “A New Simple Approach for ConstructingImplied Volatility Surfaces.” Working paper.

Derman, Emanuel, and Iraj Kani. 1994. “The Volatility Smile and Its Implied Tree.”Goldman Sachs Quantitative Strategies Research Notes, January.

Derman, Emanuel, and Iraj Kani. 1997. “Stochastic Implied Trees: Arbitrage Pric-ing With Stochastic Term and Strike Structure of Volatility.” Goldman SachsQuantitative Strategies Research Notes, April.

Dupire, Bruno. 1994. “Pricing with a Smile.” Risk 7 (1): 18–20.Gatheral, Jim. 2005. The Volatility Surface. Hoboken, NJ: John Wiley & Sons.Gurrieri, Sébastien. 2010. “A Class of Term Structures for SVI Implied Volatili-

ty”, Working paper. Available at http://ssrn.com/abstract=1779463 or http://dx.doi.org/10.2139/ssrn.1779463.

Wilmott, Paul. 2006. Paul Wilmott on Quantitative Finance, 2nd ed. Hoboken, NJ:John Wiley & Sons.

PROBLEMS

4.1 From Implied to Local Volatility

(a) Assume zero rates and dividends. Given 𝜕cBS𝜕T

= KN′(d2)𝜎2√T

and 𝜕cBS𝜕𝜎

=

K√TN′(d2) establish Equation (4.2). Hint: Consult Problem 2.2.

(b) Show that if there is no implied volatility smile—that is, σ* is only afunction of maturity—then: 1

T∫ T0 𝜎

2loc(t, St)dt = 𝜎∗2(T). Hint: Calculate

ddT

(T𝜎∗2(T)).







(c) Show that if σ* is only a linear function of strike close to at-the-money,then 𝜎loc(T,K) ≈

𝜎∗(K)1+d1K𝜎∗′(K)

√Tand thus 𝜕𝜎loc

𝜕K≈ 2𝜎∗′(K).Hint: Show that

𝜎2loc(T,K) = 𝜎∗2(K)

(1+d1K𝜎∗′(K)√T)2−𝜎∗d1

√T(K𝜎∗′(K)

√T)2

and then assume that the

term in (𝜎∗′(K))2 is negligible.

4.2 Market Price of Volatility Risk

Consider the stochastic volatility framework of (4.3) and (4.4) and define𝜆t =

𝛼t−Λt𝜔t

. Let ft = f (t, St, vt) be the value of an option on the underlyingasset S.

(a) Suppose we only delta-hedge the option. Using the Ito-Doeblin theoremshow that the change in value of the hedging portfolio is dΠt = 𝛾f,tdt +𝜕f𝜕vdvt

(b) Show that dΠt − rΠtdt = 𝜆t𝜔t𝜕f𝜕vdt + 𝜔t

𝜕f𝜕vdZt

(c) Examine the conditional expectation and standard deviation at time t ofdΠtΠt

and explain why 𝜆t is called the “market price of volatility risk.”

4.3 Local Volatility Pricing

Consider an underlying stock S currently trading at S0 = 100 that does notpay any dividend. Assume the implied volatility function is 𝜎loc(t, S) = 0.1 +0.1−0.15×ln(S∕S0)√

t, and that interest rates are zero.

(a) Produce the graph of the corresponding local volatility surface usingEquation (4.2) for spots 0 to 200 and maturities zero to five years.

(b) Write a Monte Carlo algorithm to price the following one-year payoffsusing 252 time steps and, for example, 10,000 paths:

■ “Capped quadratic” option: min(1,

S21

S20

)■ Asian at-the-money-call: max

(0, S0.25+S0.5+S0.75+S1

4× S0− 1

)■ Barrier call: max(0, S1 − S0) if S always traded above 80 using 252daily observations, 0 otherwise.


APPENDIX 4.A: DERIVATION OF DUPIRE’S EQUATION

For diffusion processes of the form:

dXt = a(t,Xt)dt + b(t,Xt)dWt

the probability density 𝜋(t,x) = ℙ{Xt = x} must solve the forwardKolmogorov equation:

𝜕𝜋𝜕t

= − 𝜕𝜕x

(a𝜋) + 12𝜕2

𝜕x2(b2𝜋)

When interest rates are zero, the risk-neutral spot price process withlocal volatility is simply:

dSt = 𝜎loc(t, St)StdWt

and the forward Kolmogorov equation at time T reduces to:

𝜕𝜋𝜕T

(T, S) = 12𝜕2

𝜕S2(S2𝜎2

loc(T, S)𝜋(T, S))

Additionally, the price of the call struck atKwith maturity T is c(K,T) =∫ ∞K (S − K)𝜋(T, S)dS. Taking the derivative with respect to T and substitutingthe forward Kolmogorov equation we get:

dcdT

= ∫∞

K(S − K) 𝜕𝜋

𝜕T(T, S)dS = 1

2∫∞

K(S − K) 𝜕

2

𝜕S2(S2𝜎2

loc(T, S)𝜋(T, S))dS

Integrating by parts we get:

dcdT

= 12

[(S − K) 𝜕

𝜕S(S2𝜎2

loc(T, S)𝜋(T, S))

]∞K− 1

2∫∞

K

𝜕𝜕S

(S2𝜎2loc(T, S)𝜋(T, S))dS

= 12K2𝜎2

loc(T,K)𝜋(T,K)

which is Dupire’s equation after recognizing the implied distribution density𝜋 = 𝜕

2c𝜕K2 . Note that we assumed all quantities taken as S → ∞ to vanish,

which is verified in all practical applications.

CHAPTER 5Volatility Derivatives

Option traders who hedge their delta have long realized that their optionbook is exposed to many other market variables, chief of which is volatility.In fact we will see that the P&L on a delta-hedged option position is drivenby the spread between two types of volatility: the instant realized volatility ofthe underlying stock or stock index, and the option’s implied volatility. Thusoption traders are specialists of volatility, and naturally they want to tradeit directly. This prompted the creation of a new generation of derivatives:forward contracts and options on volatility itself.

5-1 VOLATILITY TRADING

Delta-hedged options may be used to trade volatility, specifically the gapbetween implied volatility 𝜎* and realized volatility 𝜎—another word forhistorical volatility. To see this consider the P&L breakdown of an optionposition over a time interval Δt:

P&LΔt = 𝛿 × (ΔS) + 12Γ × (ΔS)2 + Θ × (Δt) + 𝜌 × (Δr) + × (Δ𝜎∗) + · · ·

where δ, Γ, Θ, ρ, are the option’s Greeks, ΔS is the change in underlyingspot price, Δr is the change in interest rate, Δ𝜎* is the change in impliedvolatility, and “· · ·” are high-order terms completing the Taylor expansion.

Assuming that the option is delta-hedged, the interest rate and impliedvolatility are constant, and high-order terms are negligible, we may write:

P&LΔt ≈12Γ × (ΔS)2 + Θ × (Δt)

59


From the Black-Scholes partial differential equation we get Θ ≈−1

2ΓS2𝜎∗2 where S is the initial spot price. Substituting this result and

factoring by the dollar gamma 12ΓS2 we obtain the P&L proxy equation:

P&LΔt ≈12ΓS2

[(ΔSS

)2

−(𝜎∗

√Δt

)2]

The quantity(ΔSS

)2is the squared realized return on the underlying

asset; for short Δt it may be viewed as instant realized variance. Thedelta-hedged option’s P&L is thus determined by the difference betweenrealized and implied variances multiplied by the dollar gamma. In particular,for positive dollar gamma, the P&L is positive whenever realized varianceexceeds implied variance, breaks even when both quantities are equal, andis negative whenever realized variance is lower than implied variance.

If the option is delta-hedged at N regular intervals of length Δt untilmaturity, the cumulative P&L proxy equation is then:

Cumulative P&L ≈ 12

N−1∑t=0

ΓtS2t

[(ΔStSt

)2

−(𝜎∗

√Δt

)2]

(5.1)

which captures the difference between realized and implied variancesweighted by dollar gammas over the option’s lifetime.

The continuous-time version of Equation (5.1) was derived by Carr andMadan (1998) and is exact rather than approximate:

Cumulative P&L = 12∫

T

0er(T−t)ΓtS2t

[𝜎2t − 𝜎∗2]dt (5.2)

where r is the continuous interest rate and 𝜎t is the instant realized volatilityat time t.

In practice this rather clear picture is blurred by the fact that impliedvolatility varies over time, impacting hedge ratios and thus P&L.

The cumulative P&L equations also have another interpretation: Anoption issuer is in the business of underwriting option payoffs and then repli-cating them by trading the underlying asset. In the Black-Scholes world, suchreplication is riskless, but in practice there is a mismatch given by the cumu-lative P&L equations (subject to their assumptions). At times the mismatchis negative, and at other times it is positive, resulting in a distribution ofpossible P&L outcomes. By the law of large numbers and the central-limittheorem, repeating option transactions often for a small profit narrows theP&L distribution of the option book and results in positive revenues onaverage.

Volatility Derivatives 61

5-2 VARIANCE SWAPS

Variance swaps appeared in the mid-1990s as a means to trade (the squareof) volatility directly rather than through delta-hedging. Their attractiveproperty is that they can be approximately replicated with a static portfolioof vanilla options, providing a robust fair price.

Recently the Chicago Board Options Exchange (CBOE) launcheda redesigned version of their variance futures with a quotation systemmatching over-the-counter (OTC) conventions. This interesting initiativemay signal a shift from OTC to listed markets for variance swaps.

5-2.1 Variance Swap Payoff

From the buyer’s viewpoint, the payoff of a variance swap on an underlyingS with strike Kvar and maturity T is:

Variance Swap Payoff = 10,000 × (𝜎2Realized

− K2var)

where 𝜎Realized is the annualized realized volatility of the N daily log-returnson S between t = 0 and t = T under zero-mean assumption:

𝜎Realized =

√√√√252N

N−1∑i=0

ln2Si+1Si

For example, a one-year variance swap on the S&P 500 struck at 25%pays off 10,000 × (0.262 – 0.252) = $51 if realized volatility ends up at26%. Figure 5.1 shows the variance swap payoff as a function of realizedvolatility. We can see that the shape is convex quadratic, which implies that

Payoff

–10,000 × K2var

σRealizedKvar

FIGURE 5.1 Variance swap payoff as a function ofrealized volatility.


profits are amplified and losses are discounted as realized volatility goesaway from the strike.

In practice the quantity of variance swaps is often specified in “veganotional,” for example, $100,000. This means that, close to the strike,each realized volatility point in excess of the strike pays off approximately$100,000. This is achieved by setting the actual quantity, called variancenotional or variance units as:

Variance units =Vega notional200 × Kvar

For example, with a 25% strike and $100,000 vega notional, the num-ber of variance units is simply 2,000. If realized volatility is 26% the payoffis 2,000 × [$10,000 × (.262 – .252)] = $102,000 which is close to the veganotional as required.

FOCUS ON VARIANCE FUTURES

The new variance futures launched by CBOE in December 2012 allowinvestors to trade variance swaps in a standardized format with all thebenefits of listed securities: price transparency, market-making require-ments, liquidity.

The futures are available in 1-, 2-, 3-, 6-, 9-, and 12-monthrolling maturities. The strike is set the day before the start date(typically the third Friday of the month) using a VIX-type calculation(see Section 5-4).

Trading conventions were adjusted to reflect OTC market prac-tices, in particular:

■ Prices are quoted in terms of implied volatility, which may becompared against the fair value of future variance observeduntil maturity.

■ The quantity is set to $1,000 vega notional, which is convertedinto variance units at trade execution.

■ On the expiration date, the exchange will credit or debit thedifference between realized variance and the square of strike,adjusted to cancel the effects of the daily variation margin.

5-2.2 Variance Swap Market

Variance swaps trade mostly over the counter. Table 5.1 shows mid-marketprices for various underlyings and maturities.


TABLE 5.1 Variance Swap Mid-Market Prices on ThreeStock Indexes as of August 21, 2013

Maturity S&P 500 EuroStoxx 50 Nikkei 225

1 month 15.23 19.37 27.583 months 16.50 20.89 28.356 months 18.01 22.22 27.5512 months 19.59 23.88 26.9818 months 20.43 24.50 26.2224 months 21.20 25.12 26.16

5-2.3 Variance Swap Hedging and Pricing

As stated earlier, variance swaps can be replicated using a static portfolio ofcalls and puts of the same maturity, which must then be delta-hedged. Tosee this, we must explicate the connection between variance swaps and thelog-contract whose payoff is –ln ST/F where F is the forward price.

Applying the Ito-Doeblin theorem to ln St yields:

lnSTS0

= ∫T

0

1StdSt −

12∫

T

0

1S2t

(dSt)2 = ∫T

0

1StdSt −

12∫

T

0𝜎2t dt

where 𝜎t is the underlying asset’s instant volatility, which may be stochas-tic. Rearranging terms, we can see that realized variance may be replicatedby continuously maintaining a position of 2/St in the underlying stock andholding onto a certain quantity of log-contracts:

∫T

0𝜎2t dt = 2∫

T

0

1StdSt − 2 ln

STS0

Assuming that the risk-neutral dynamics of the underlying price are ofthe form dSt = 𝜈tStdt + 𝜎tStdWt where 𝜈t is the risk-neutral drift, we have

𝔼

(∫

T

0

1StdSt

)= 𝔼

(∫

T

0𝜈tdt

)= ln F

S0and thus the fair value of variance

matches the fair value of two log-contracts:

𝔼

(∫

T

0𝜎2t dt

)= 𝔼

(−2 ln

STF

)The log-contract does not trade, but from Section 3-2 we know that any

European payoff can be decomposed as a portfolio of calls and puts struckalong a continuum of strikes. Specifically we have the identity:

− lnSTF

= 1 −STF

+ ∫F

0

1K2

max(0,K − ST)dK + ∫∞

F

1K2

max(0, ST − K)dK


In other words the log-contract may be replicated with a portfoliowhich is:

■ Short a forward contract on S struck at the forward price F■ Long all puts struck at K < F in quantities dK / K2

■ Long all calls struck at K > F in quantities dK / K2

The corresponding fair value of annualized variance is then:

K2var =

2T𝔼(− ln

STF

)= 2erT

T

[∫

F

0

1K2

p (K)dK + ∫∞

F

1K2

c(K)dK]

where r is the interest rate for maturity T, p(K) is the price of a put struckat K, and c(K) is the price of a call struck at K.

In practice only a finite number of strikes are available, and we have theproxy formula:

K2var ≈

2erT

T

[n∑i=1

p(Ki

)K2i

ΔKi +n+m∑i=n+1

c(Ki)K2i

ΔKi

], (5.3)

where K1 < · · · < Kn ≤ F ≤ Kn+1 < · · · < Kn+m are the successive strikesof n puts worth p(Ki) and m calls worth c(Ki), and ΔKi = Ki – Ki–1 is thestrike step.

A more accurate calculation based on overhedging of the log-contractis discussed in Demeterfi, Derman, Kamal, and Zhou (1999). An alternativederivation of the hedging portfolio and price based on the property thatvariance swaps have constant dollar gamma can be found in Bossu, Strasser,and Guichard (2005).

5-2.4 Forward Variance

Two variance swaps with maturitiesT1 <T2 may be combined to capture theforward variance observed between T1 and T2. Indeed under the zero-meanassumption variance is additive and thus:

T1𝜎2Realized

(0,T1) + (T2 − T1)𝜎2Realized(T1,T2) = T2𝜎2Realized

(0,T2)

where 𝜎2Realized

(s, t) denotes realized volatility between times s and t. Thus:

𝜎2Realized

(T1,T2) =T2

T2 − T1𝜎2Realized

(0,T2) −T1

T2 − T1𝜎2Realized

(0,T1)


This implies that the fair strike of a forward variance swap is:

Kvar(T1,T2) =

√T2

T2 − T1K2var(0,T2) −

T1

T2 − T1K2var(0,T1)

and, when interest rates are zero, the corresponding hedge is long T2T2−T1

variance swaps maturing at T2 and short T1T2−T1

variance swaps maturingat T1.

When interest rates are non-zero, the quantity of the short leg should beadjusted to T1

T2−T1e−r(T2−T1) because the near-term variance swap payoff at

time T1 will carry interest r between T1 and T2.

5-3 REALIZED VOLATILITY DERIVATIVES

Variance swaps are the simplest kind of realized volatility derivatives; that is,derivative contracts whose payoff is a function f (𝜎Realized) of realized volatil-ity. Other examples include:

■ Volatility swaps, with payoff 𝜎Realized − Kvol

■ Variance calls, with payoff max(0, 𝜎2

Realized− K

)■ Variance puts, with payoff max

(0,K − 𝜎2

Realized

)Volatility swaps are offered by some banks to investors who prefer them

to variance swaps. Deep out-of-the-money variance calls are often embeddedin variance swap contracts on single stocks in the form of a cap on realizedvariance; this is because variance could “explode” when, for instance, theunderlying stock is approaching bankruptcy.

Note that the volatility swap strike Kvol must be less than the varianceswap strike Kvar under penalty of arbitrage. This property may also be seenthrough Jensen’s inequality:

Kvol = 𝔼(√𝜎2Realized

)≤

√𝔼(𝜎2Realized

)= Kvar

The ratio KvarKvol

≥ 1 is called the convexity adjustment by practitioners.

Valuing realized volatility derivatives requires a model. Since realizedvariance is tradable, a natural idea here is to think of it as an underlyingasset and apply standard option valuation theory.


However an adjustment is necessary to account for the fact that, as atradable asset, realized variance is a mixture of past variance and futurevariance; specifically:

vt =tT𝜎2(0, t) + T − t

TK2var(t,T) (5.4)

where vt is the forward price of variance started at time 0 and ending at timeT, 𝜎(0,t) is the historical volatility observed over [0, t] and Kvar(t,T) is thefair strike at time t of a new variance swap expiring at time T.

Therefore we cannot assume that the volatility of vt is constant as in theBlack-Scholes model. Instead we may assume forward-neutral dynamics ofthe form:

dvt∕vt = 2𝜔T − tT

dWt

where 𝜔 is a volatility of (future) volatility parameter and 2 is a conversionfactor from volatility to variance.1 In this fashion the diffusion coefficient2𝜔T−t

Tlinearly collapses to zero as we approach maturity, as suggested by

Equation (5.4).This simple modeling approach allows us to find closed-form formulas

for the price of many realized volatility derivatives. For example, the fairstrike of a volatility swap can be shown to be:

Kvol = Kvar exp(−16𝜔2T

)and thus the convexity adjustment is simply Kvar

Kvol= exp

(16𝜔2T

).

Given Kvol and Kvar from the market we may then estimate ω:

𝜔 =

√6T

lnKvar

Kvol

In practice volatility swaps are illiquid but, as suggested by Carr andLee (2009), the fair value of realized volatility may be approximated withat-the-money implied volatility. For example, if one-year fair varianceis 30% and one-year at-the-money implied volatility is 28%, we find

𝜔 ≈√

6 ln 30%28% = 64.3%.

1Applying the Ito-Doeblin theorem to√vt we indeed obtain dynamics of the form

d√vt∕

√vt = · · ·dt + 𝜔T−t

TdWt.


5-4 IMPLIED VOLATILITY DERIVATIVES

Implied volatility derivatives are derivative contracts whose payoff is afunction f (𝜎Implied) of implied volatility. There are many choices for impliedvolatility and the most popular one is the Volatility Index (VIX) calculatedby CBOE, which is the short-term fair variance derived from listed optionson the S&P 500 in accordance with Equation (5.3).

The VIX is not a tradable asset, but CBOE nevertheless developed VIXfutures and VIX options, which have become increasingly popular.

Specifically, the VIX is a pro rata temporis average of two subindexesbased on front- and next-month2 listed options in order to reflect the 30-dayexpected volatility. Each subindex is calculated according to the generalizedformula: (

VIX100

)2

= 2erT

T

∑i

ΔKi

K2i

Q(Ki) −1T

(FK0

− 1)

where:

■ T is the maturity■ r is the continuous interest rate for maturity T■ F is the forward price of the S&P 500 derived from listed option prices■ K0 is the first strike below F■ Ki is the strike price of the |i|th out-of-the money listed option: a call ifi > 0 and a put if i < 0

■ ΔKi is the interval between strike prices measured as 12(Ki+1 − Ki−1)

■ Q(Ki) is the mid-price of the listed call or put struck at Ki.

There are many other practical details that can be found in the CBOEwhite paper on VIX (2009), such as how exactly the forward price is calcu-lated or how the constituent options are selected.

5-4.1 VIX Futures

VIX futures are futures contracts on the VIX. On the settlement date T thefollowing dollar amount is credited or debited by the exchange:

1,000 × (VIXT − Futt)

where 1,000 is the multiplication factor, VIXT is the settlement level3 of theVIX, and Futt is the trading price at time t.

2Or the second- and third-month listed options if front-month options expire in lessthan one week.3Note that VIX futures do not settle the VIX itself but a special opening quotationof the VIX determined by automated auction of the constituent options prior to theopening of trading.


0

5

10

15

20

25

30

Spot Oct 12 Nov 12 Dec 12 Jan 13 Feb 13 Mar 13 Apr 13 May 13 Jun 13

FIGURE 5.2 Term structure of VIX futures as of September 26, 2012.Source: Bloomberg.

Figure 5.2 shows the term structure of VIX futures as of September 26,2012. We can see that the curve is upward sloping; however, at times itmay be downward sloping because VIX futures, contrary to stock or equityindex futures, are not constrained by any carry arbitrage since the VIX isnot a tradable asset.

It is worth emphasizing that VIX futures are bets on the future levelof implied volatility, which itself is the market’s guess at subsequentrealized volatility. In this sense VIX futures are forward contracts onforward realized volatility—a potentially difficult level of “forwardness” todeal with.

Close to expiration, the futures tend to be highly correlated with theVIX. When futures expire in a month, they have about 0.5 correlation withthe VIX. Futures that expire in five months or more exhibit almost no cor-relation.

5-4.2 VIX Options

VIX options are vanilla calls and puts on the VIX. On the settlement date Tthe following dollar amount is credited or debited by the exchange:

■ For a call: 100 ×max (0,VIXT − K)■ For a put: 100 ×max (0,K − VIXT)


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

1m 2m 3m 4m 5m 6m

FIGURE 5.3 Implied volatility term structure of at-the-money VIX options as ofSeptember 26, 2012.Source: Bloomberg.

where 100 is the multiplication factor, VIXT is the settlement level4 of theVIX, and K is the strike level.

For lack of a better consensus model, VIX option prices are commonlyanalyzed through Black’s model to compute VIX implied volatilities.Figure 5.3 shows the term structure of at-the-money VIX implied volatility.We can see that the shape is downward sloping: short-term VIX futures areexpected to be more volatile than long-term ones.

Similar to VIX futures, VIX option prices tend to have high correla-tion with the VIX close to expiration and low correlation far away fromexpiration.

Synthetic put-call parity holds for VIX options, which entails that VIXimplied volatility is the same for calls and puts. Specifically:

VIX Call Price − VIX Put Price = Discount Factor

× (VIX Future Price − Strike)

where Discount Factor is the price of $1 paid on the expiration date.

4Note that VIX options do not settle the VIX itself but a special opening quotationof the VIX determined by automated auction of the constituent options prior to theopening of trading.


REFERENCES

Bossu, Sébastien. 2005. “Arbitrage Pricing of Equity Correlation Swaps.” JPMorganEquity Derivatives report.

Bossu, Sébastien, Eva Strasser, and Régis Guichard. 2005. “Just What You Need toKnow about Variance Swaps.” JPMorgan Equity Derivatives report.

Carr, Peter, and Roger Lee. 2009. “Volatility Derivatives.” Annual Review of Finan-cial Economics 1: 1–21.

Carr, Peter, and Dilip Madan. 1998. “Towards a Theory of Volatility Trading.”In Volatility: New Estimation Techniques for Pricing Derivatives, edited byR. Jarrow, 417–427. London: Risk Books.

“The CBOE Volatility Index—VIX.” 2009. CBOE White Paper.Demeterfi, Kresimir, Emanuel Derman, Michael Kamal, and Joseph Zhou. 1999.

“More than You EverWanted to Know about Volatility Swaps.” Goldman SachsQuantitative Strategies Research Notes, March.

PROBLEMS

5.1 Delta-Hedging P&L Simulation

Consider a long position in 10,000 one-year at-the-money calls on ABCInc.’s stock currently trading at $100. Assume zero interest rates and div-idends and 30% implied volatility.

(a) Suppose the “real” stock price process is a geometric Brownian motionwith 5% drift and 29% realized volatility. Simulate the evolution of thecumulative delta-hedging P&Lwith 252 time steps using Equation (5.1).Show one path where the final P&L is positive and one path where it isnegative.

(b) Using 10,000 simulations, compute the empirical distribution of the finalcumulative delta-hedging P&L. What is the average P&L?

5.2 Volatility Trading with Options

(a) You are long a call, which you delta-hedge continuously until maturity.Your cumulative P&L is given by Equation (5.2).i. Suppose the “real” stock price process is a geometric Brownian

motion with drift 𝜇 and realized volatility 𝜎 > 𝜎*. Show that you areguaranteed a positive profit.

ii. Suppose the “real” stock price process is dSt∕St = 𝜇tdt + 𝜎tdWtwhere 𝜇t and 𝜎t are stochastic with 𝔼t(𝜎2t ) > 𝜎∗2. Show that yourexpected P&L (unconditionally from time 0) is positive.


(b) You are long two one-year calls in quantities q1 and q2 on two indepen-dent underlying stocks, which you delta-hedge continuously until matu-rity. Suppose that the distributions of the two cumulative P&L equationsare normal with means m1, m2, and standard deviations s1, s2. What isthe distribution of the aggregate cumulative P&L of your option book?Show that its standard deviation must be less than (q1 + q2)max(s1, s2).

5.3 Fair Variance Swap Strike

Using your SVI calibration results from Problem 2.5 and assuming zero inter-est rates, estimate the corresponding fair variance swap strike in accordancewith Equation (5.3).

5.4 Generalized Variance Swaps

A generalized variance swap has payoff:

10,000 ×

(252N

N−1∑i=0

f(Si)ln2

Si+1Si

− K2gvar

)

where f(S) is a general function of the underlying spot price S, St is the spotprice at time t, N is the number of trading days until maturity, and Kgvar isthe strike level. Assume zero interest and dividend rates, and that dSt∕St =𝜎tdWt under the risk-neutral measure.

(a) Let g(S) = ∫S

1dy∫

y

1

f (x)x2

dx. Using the Ito-Doeblin theorem, show that:

∫T

0f (St)𝜎2t dt = 2g(ST) − 2g(S0) − 2∫

T

0g′(St)dSt

What can you infer about the fair value and hedging strategy of gener-alized variance?

(b) Show that the fair strike is

Kgvar =

√√√√ 2T

(∫

S0

0

f (K)K2

p(K)dK + ∫∞

S0

f (K)K2

c(K)dK

)

(c) Application: corridor variance swap. What does the formula for the fairstrike become in case of the corridor variance swap where daily realizedvariance only accrues at time t when L < St < U?


5.5 Call on Realized Variance

Consider the model dvt∕vt = 2𝜔T−tTdWt for the forward price at time t of

realized variance observed over a fixed period [0,T]. In particular, vT is theterminal historical variance, and v0 = 𝔼(vT) is the undiscounted fair strikeof a variance swap at time 0.

(a) Show that the price of an at-the-money forward call on realized variancewith payoff max(0, vT − v0) is given as:

Varcall0 = e−rTv0

[2N

(𝜔

√T3

)− 1

]

(b) Using a first-order Taylor expansion show that

Varcall0 ≈ 2√2𝜋

e−rTv0𝜔

√T3

CHAPTER 6Introducing Correlation

Correlation is almost as ubiquitous as volatility in quantitative finance.For example the downward-sloping volatility smile observed in equitiesmay be explained by the negative correlation between stock prices andvolatility. In this chapter we introduce various measures of correlationbetween assets, investigate their properties, and present simple multiassetextensions of the Black-Scholes and Local Volatility models.

6-1 MEASURING CORRELATION

Correlation is the degree to which two quantities are linearly associated.A correlation of +1 or −1 means that the linear relationship is perfect, whilea correlation of 0 typically1 indicates independence.

There are two kinds of correlation between two financial assets:

1. Historical correlation, based on historical returns;2. Implied correlation, derived from option prices.

6-1.1 Historical Correlation

Historical correlation between two assets S(1) and S(2) is usually measuredas the Pearson’s correlation coefficient between their N historical returnsobserved at regular intervals:

𝜌†1,2 =Cov†1,2

𝜎†1𝜎†2

=

N∑i=1

(r(1)i − r(1)

)(r(2)i − r(2)

)√√√√ N∑

i=1

(r(1)i − r(1)

)2×

N∑i=1

(r(2)i − r(2)

)2

1Recall that if two random variables X, Y are independent their correlation must bezero; however, the converse is not necessarily true.

73


–0.4

–0.2

0

Jan-00 May-01 Sep-02 Feb-04 Jun-05 Nov-06 Mar-08 Aug-09 Dec-10 Apr-12

0.2

0.4

0.6

0.8

1

FIGURE 6.1 Historical correlation of daily returns between Apple and Microsoftover a three-month rolling window since 2000.

where Cov†1,2 is historical covariance, 𝜎†’s are historical standard devia-

tions, r(j)i is the return on asset S(j) for observation i, and r(j) = 1N

∑Ni=1 r

(j)i is

the mean return on asset S(j). Returns may be computed on an arithmetic orlogarithmic basis; occasionally the mean returns are assumed to be zero.

Figure 6.1 shows the evolution of the historical correlation betweenMicrosoft and Apple over a three-month rolling window since 2000. Wecan see that this correlation has varied quite significantly over time.

Note that using daily returns can produce misleading results for assetstrading within different time zones; in this case it is preferable to estimatecorrelation using weekly returns. Figure 6.2 compares the two methodsfor the S&P 500 and Nikkei 225 indexes. We can see that the correlationobserved on weekly returns is significantly higher.

6-1.2 Implied Correlation

Implied correlation between two assets S(1) and S(2) is derived from an optionprice, such as a quote for an over-the-counter (OTC) basket option. Typicallythe quote is converted into an implied basket volatility 𝜎∗

Basketfrom which

implied correlation may be extracted through the formula:

𝜎∗Basket

=√w2

1𝜎∗21 +w2

2𝜎∗22 + 2w1w2𝜎

∗1𝜎

∗2𝜌

∗1,2, that is,

𝜌∗1,2 =𝜎∗2Basket

−w21𝜎

∗21 −w2

2𝜎∗22

2w1w2𝜎∗1𝜎

∗2

=𝜎∗2Basket

− (w21𝜎

∗21 +w2

2𝜎∗22 )

(w1𝜎∗1 +w2𝜎

∗2)

2 − (w21𝜎

∗21 +w2

2𝜎∗22 )

wherewj is the weight on asset S(j) and 𝜎∗j is the implied volatility of asset S(j).

Introducing Correlation 75

–0.4Jan-00 May-01 Sep-02 Feb-04 Jun-05 Nov-06 Mar-08 Aug-09 Dec-10 Apr-12

–0.2

0

0.2

0.4

0.6

0.8

SPX-NKY Daily Returns SPX-NKY Weekly Returns1

FIGURE 6.2 Historical correlation of daily and weekly returns between S&P 500and Nikkei 225 over a three-month rolling window since 2000.

Conventionally all implied volatilities are for the same moneyness levelk (strike over spot) and maturity T, and weights are equal.

6-2 CORRELATION MATRICES

Very often we are interested in correlation for a selection of n ≥ 2 assets.This leads to a correlation matrix of the form:

R =

⎛⎜⎜⎜⎜⎜⎜⎝

1 𝜌1,2 𝜌1,3 · · · 𝜌1,n𝜌2,1 1 𝜌2,3 · · · 𝜌2,n𝜌3,1 𝜌3,2 1 · · · 𝜌3,n⋮ ⋮ ⋮ ⋱ ⋮

𝜌n,1 𝜌n,2 𝜌n,3 · · · 1

⎞⎟⎟⎟⎟⎟⎟⎠where 𝜌i,j is the pairwise correlation coefficient between assets S(i) and S(j),which may either be historical or implied. Note that R is symmetric because𝜌i,j = 𝜌j,i.

Not every symmetric matrix with entries in [−1, 1] and a diagonal of1’s is a candidate for a correlation matrix R. This is because the correlationbetween assets S(i) and S(j) and assets S(j) and S(k) says something about thecorrelation between assets S(i) and S(k)—intuitively, if Microsoft and Appleare highly correlated, and Apple and IBM are also highly correlated, thenMicrosoft and IBM must also have some positive correlation.


–1.0

–1.0

–0.5

–0.5

0.0

0.0

0.5

0.5

1.0

1.0–1.0

–0.5

0.0

0.5

1.0

FIGURE 6.3 Envelope of admissible correlation values when n = 3.

Figure 6.3 shows the envelope of admissible correlation values whenn = 3. We can see that certain regions, such as around the corner (−1, −1,−1) are not admissible.

Specifically, correlation matrices must be positive-semidefinite; that is,their eigenvalues must all be nonnegative. This property is always verifiedfor historical correlation but not necessarily for implied correlation. Addi-tionally the sum of all eigenvalues must equal the trace, that is, n.

A common fix for an indefinite candidate matrixM is to replace its neg-ative eigenvalues with zeros and adjust its positive eigenvalues to maintaina sum of n:

R = ΩDadjΩT

where Ω is the orthogonal matrix of eigenvectors of M with eigenvalues(λ1,… , λn) and Dadj is the diagonal matrix of adjusted eigenvalues with

entries 𝜆adji =𝜆+i∑nj=1 𝜆

+j. Alternatively one may use the method proposed by

Higham (2002).In equities correlation matrices have other empirical properties. Plerou

et al. (2002) and Potters, Bouchaud, and Laloux (2005) found for U.S. stocks


that the top eigenvalue typically dominates all the other ones. Furthermore,the corresponding eigenvector is more or less an equally weighted portfolioof all the stocks. This suggests that one factor (“the market”) strongly drivesthe behavior of each stock.

6-3 CORRELATION AVERAGE

To summarize the overall level of correlation across n assets, it is commonpractice to compute the average of the correlation matrix, excluding thediagonal of 1’s. The formula for a given weighting vector x is then:

𝜌(x) =

∑i<j

xixj𝜌i,j∑i<j

xixj= xTRx − xTx

(xTe)2 − xTx(6.1)

where e is the vector of 1’s. In the main case of interest where all the weightsare nonnegative we have:

−1 ≤ − xTx(xTe)2 − xTx

≤ 𝜌(x) ≤ (n − 1) xTx(xTe)2 − xTx

≤ 1

but in general 𝜌(x) could lie outside of these bounds. In practice, when apply-ing sensible weights to a large equity correlation matrix, 𝜌(x) can safely beassumed to be positive.

Common choices for x are:

■ Equal weights: x = e. In this case the average correlation formulasimplifies to:

𝜌(e) = 2n(n − 1)

∑i<j

𝜌i,j

and we have the bounds:

− 1n − 1

≤ 𝜌(e) ≤ 1

■ Market capitalization weights: x = w. This is particularly relevantwhen the n stocks are the constituents of an equity index such as theS&P 500.


■ Volatility and market capitalization weights: x = (w1𝜎1,… ,wn𝜎n)T ,where 𝜎’s may either be historical or implied volatilities. This case isparticularly appealing because of the identity2 or shortcut formula:

𝜌⎛⎜⎜⎝w1𝜎1⋮

wn𝜎n

⎞⎟⎟⎠ =𝜎2Basket

−n∑i=1

w2i 𝜎

2i(

n∑i=1

wi𝜎i

)2

−n∑i=1

w2i 𝜎

2i

where 𝜎Basket is the volatility of the all-stock portfolio with weights w.Assimilating an equity index to a portfolio of stocks with fixed weights,3

this formula allows us to compute the average implied correlation usingonly listed option prices.

In practice, for large baskets (n > 30), these various choices for x tendto produce similar results within a few correlation points, as observed byTierens and Anadu (2004) and illustrated in Figure 6.4.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ju

l-0

2

Fe

b-0

3

Se

p-0

3

Ap

r-0

4

No

v-0

4

Ju

n-0

5

Ja

n-0

6

Au

g-0

6

Ma

r-0

7

Oct-

07

Ma

y-0

8

De

c-0

8

Ju

l-0

9

Fe

b-1

0

Se

p-1

0

Ap

r-1

1

No

v-1

1

Ju

n-1

2

Ja

n-1

3

Market cap. Market cap. and vol.

FIGURE 6.4 Realized average correlation for the EuroStoxx 50 index over asix-month rolling window using market capitalization weights and volatility andmarket capitalization weights.

2Note that the identity is exact for arithmetic returns but only approximate for log-arithmic returns.3Note that in reality equity index weights continuously change with stock prices.However, these variations tend to be limited, especially over short time horizons.


6-3.1 Correlation Proxy

Equation (6.1) is related to a mathematical quantity known as the Rayleighquotient ℜ(x) = xTRx

xTx; specifically, dividing both numerator and denomina-

tor by nxTx = (xTx)(eTe):

𝜌(x) =

1nℜ(x) − 1

n(xTe)2

(xTx)(eTe)− 1n

=

1nℜ(x) − 1

n

cos2𝜃 − 1n

where θ is the angle between vectors x and e.As n → ∞ we have the proxy formula:

𝜌(x) ∼

1nℜ(x)

cos2𝜃= xTRx

(xTe)2

subject to certain technical conditions, which are met in practice. In par-ticular, for volatility and market capitalization weights, the proxy formulaequates the now well-known squared ratio of basket volatility to averagestock volatility:

𝜌(x) ∼⎛⎜⎜⎜⎝𝜎Basket∑i

wi𝜎i

⎞⎟⎟⎟⎠2

FOCUS ON THE PROXY FORMULA

It is easy to establish the proxy formula when all correlation coeffi-cients are positive (see, e.g., Bossu and Henrotte (2012)). However,when some correlation coefficients are negative we must use a moreelaborate proof. Specifically, using the spectral decomposition of R,we may write:

ℜ(x) =n∑i=1𝜆i

(xTvi)2

(xTx)(vTi vi)

where v’s form an orthogonal basis of eigenvectors and 𝜆’s are theirassociated eigenvalues.


Therefore ℜ(x) ≥ n∑i=1𝜆imin

1≤j≤n(xTvj)2

(xTx)(vTj vj)= nmin

1≤j≤n(xTvj)2

(xTx)(vTj vj)since all

eigenvalues must sum to n. Assuming that x is never orthogonalto any eigenvector vi (also in the limit) then ℜ(x) → ∞ and thus1nℜ(x) − 1

n∼ 1

nℜ(x). Furthermore, if x is also never orthogonal to e

(also in the limit) then cos2𝜃 − 1n∼ cos2𝜃, which completes the proof

that 𝜌(x) ∼1nℜ(x)

cos2𝜃= xTRx

(xTe)2 .

6-3.2 Some Properties of the Correlation Proxy

We now focus on some fundamental properties of the proxy formula �̂�(x) =xTRx(xTe)2 . In what follows it is assumed that the eigenvalues of R are sorted byascending order.

First, a property of the Rayleigh quotient is that it must be comprisedbetween the top and bottom eigenvalues, which implies that:

0 ≤ 𝜆1∕ncos2𝜃

≤ �̂�(x) ≤ 𝜆n∕ncos2𝜃

≤ 1cos2𝜃

Note that the lower bound 𝜆1∕ncos2𝜃

can be slightly improved in the uncon-strained case (see Problem 6.1) and that tighter numerical bounds can becomputed through quadratic optimization methods in the constrained casewhere x ≥ 0.

Second, another quantity of interest is the distance between two averagecorrelation measuresΔ = |�̂�(x) − �̂�(y)|. Restricting ourselves to vectors x andy such that xTe = yTe = 1 we may rewrite without loss of generality:

Δ = |||xTRx − yTRy||| = |||(x + y)TR(x − y)|||The Cauchy-Schwarz inequality then gives the general upper bound Δ ≤

𝜆n‖x + y‖‖x − y‖ but in practice it is not satisfactory. To find a better upperbound we must look at the spectral decomposition of R:

R =n∑i=1𝜆iviv

Ti

vTi vi


where vi is an eigenvector with associated eigenvalue 𝜆i. Thus, for any vec-tors a and b:

aTRb√aTa bTb

=n∑i=1𝜆i

aTvi√aTa vTi vi

vTi b√vTi vi b

Tb=

n∑i=1𝜆i cos (̂a, vi) cos (̂vi,b)

where (̂u, v) denotes the absolute angle in [0, π] between any two vectors uand v.

Recalling that the top eigenvalue of stock correlation matrices domi-nates all other eigenvalues, we are induced to split the sum accordingly:

aTRb√aTa bTb

=n−1∑i=1𝜆i cos (̂a, vi) cos (̂vi,b) + 𝜆n cos (̂a, vn) cos (̂vn,b)

Furthermore cos 𝛼 cos 𝛽 = cos(𝛼 + 𝛽) + sin 𝛼 sin 𝛽, so that:

aTRb√aTa bTb

=n−1∑i=1𝜆i cos (̂a, vi) cos (̂vi,b) + 𝜆n sin (̂a, vn) sin (̂vn,b)

+ 𝜆n cos[(̂a, vn) + (̂vn,b)]

We now invoke the fifth property of the Euclidean metric4 to get|cos (̂a, vi)| ≤ sin (̂a, vn), |cos (̂vi,b)| ≤ sin (̂vn,b) and for (̂a, vn) + (̂vn,b) ≤ 𝜋2 :0 ≤ cos[(̂a, vn) + (̂vn,b)] ≤ cos (̂a,b), so that:|||||| aTRb√

aTa bTb

|||||| ≤ n sin (̂a, vn) sin (̂vn,b) + 𝜆n cos (̂a,b)

because the eigenvalues sum to n.Taking a = x + y, b = x − y and rearranging terms we get:

Δ ≤ n‖x + y‖‖x − y‖ [sin ̂

(x + y, vn

)sin ̂(vn,x − y) +

𝜆nn

cos ̂(x + y,x − y)]

4See Dattorro (2008) who cites Blumenthal (1933). See also Laurence et al. (2008)who cite De Finetti (1937).


In practice the quantity between brackets is usually small because x, yare “close” to vn and x + y, x − y are nearly orthogonal.

FOCUS ON THE FIFTH PROPERTY

The fifth property of the Euclidean metric is a triangle inequality forangles in three dimensions, which is surprisingly not documented inmainstream geometry textbooks. Specifically it states that for any threevectors u, v, and w we have:

|(̂u, v) − (̂v,w)| ≤ (̂u,w) ≤ (̂u, v) + (̂v,w)

where all angles are measured between 0 and 𝜋. Taking cosines weequivalently have:

cos[(̂u, v) + (̂v,w)] ≤ cos (̂u,w) ≤ cos[(̂u, v) − (̂v,w)].

As a corollary if e.g. (̂v,w) = 𝜋∕2 then |cos (̂u,w)| ≤ cos(𝜋2− (̂u, v)

)= sin (̂u, v)

6-4 BLACK-SCHOLES WITH CONSTANT CORRELATION

Extending Black-Scholes to a basket of n underlying assets S(1),… , S(n) withconstant correlation is fairly straightforward, except perhaps notation-wise.

Given a vector of volatilities (σ1,… , σn) and a correlation matrix (ρi,j),assume that the prices of the underlying assets follow n correlated geometricBrownian motions:

dS(1)t = 𝜇1S(1)t dt + 𝜎1S

(1)t dW(1)

t

dS(2)t = 𝜇2S(2)t dt + 𝜎2S

(2)t dW(2)

t

⋮

dS(n)t = 𝜇nS(n)t dt + 𝜎nS

(n)t dW(n)

t


where dW(i)t dW

(j)t ≡ 𝜌i,jdt. If the derivative’s value only depends on time

and the n spot prices, we have Dt = f (t, S(1)t ,… , S(n)t ) and we can apply the

multidimensional version of the Ito-Doeblin theorem to get:

dDt = df =𝜕f𝜕tdt +

n∑i=1

𝜕f

𝜕S(i)dS(i)t + 1

2

n∑i=1

n∑j=1

𝜕2f

𝜕S(i)𝜕S(j)𝜎i𝜎j𝜌i,jS

(i)t S

(j)t dt

=𝜕f𝜕tdt + ∇f TdSt +

12dSTt ∇

2f dSt

where ∇f and ∇2f are the gradient and Hessian of f, respectively.A portfolio long one unit of derivative and short 𝛿i =

𝜕f𝜕S(i)

units of eachasset S(i) is then riskless, and by the same reasoning as in the single-asset casewe obtain a multidimensional partial differential equation for f whose onlyparameters are the interest rate r, the volatility vector and the correlationmatrix:

rf =𝜕f𝜕t

+ rn∑i=1

𝜕f

𝜕S(i)S(i)t + 1

2

n∑i=1

n∑j=1

𝜕2f

𝜕S(i)𝜕S(j)𝜎i𝜎j𝜌i,jS

(i)t S

(j)t

Solving partial differential equations in high dimension is very hardmathematically and computationally. In practice, the numerical method ofchoice to implement the multiasset Black-Scholes model is Monte Carlo sim-ulation under the risk-neutral measure. The Cholesky decomposition of thecorrelation matrix is then typically used to generate correlated Brownianmotions from uncorrelated ones.

FOCUS ON THE CHOLESKY DECOMPOSITION

The Cholesky decomposition of a symmetric, positive-definite matrixA is the lower triangular matrixCwith strictly positive diagonal entriessuch that A = CCT. It can be computed with a short algorithm of com-plexity O(n3).

The Cholesky decomposition C of a correlation matrix R may beused to generate correlated standard normals Y = XCT from a sam-ple X of uncorrelated ones with m rows and n columns. Indeed thecovariance estimate for Y up to a multiplicative factor is:

YTY = CXTXCT ≈ CCT = R

where we used XTX ≈ I, which is true for large m.


6-5 LOCAL VOLATILITY WITHCONSTANT CORRELATION

Another straightforward extension of a popular model is local volatilitywith constant correlation (LVCC). Keeping the notations of Section 6-4, thismodel assumes dynamics of the form:

dS(1)t = 𝜇1S(1)t dt + 𝜎loc1 (t, S(1)t )S(1)t dW(1)

t

dS(2)t = 𝜇2S(2)t dt + 𝜎loc2 (t, S(2)t )S(2)t dW(2)

t

⋮

dS(n)t = 𝜇nS(n)t dt + 𝜎locn (t, S(n)t )S(n)t dW(n)

t

where 𝜎loci (t, S) is the local volatility function for asset S(i) (see Chapter 4)

and dW(i)t dW

(j)t ≡ 𝜌i,jdt as before.

The same reasoning as in Section 6-4 then applies, with identical resultsafter substituting local volatilities. Again Monte Carlo simulations are over-whelmingly preferred to other numerical methods such as multidimensionalbinomial trees or finite difference lattices.

Until recently the local volatility model with constant correlation waswidely used to price a broad range of multiasset exotic options. In Chapter 8,we introduce the next generation of models where correlation is allowedto vary.

REFERENCES

Blumenthal, Leonard M. 1933. “On the four-point property.” Bulletin of the Amer-ican Mathematical Society, 39: 423–426.

Bossu, Sébastien, and Philippe Henrotte. 2012. An Introduction to Equity Deriva-tives: Theory and Practice, 2nd ed. Chichester, UK: John Wiley & Sons.

Dattorro, Jon. 2008. Convex Optimization & Euclidean Distance Geometry. PaloAlto, CA: Meboo Publishing USA.

De Finetti, Bruno. 1937. “A proposito di correlazione.” Supplemento Statistico aiNuovi problemi di Politica, Storia ed Economia, 3: 41–57. English translationin Laurence et al. (2008).

Higham, Nicholas J. 2002. “Computing the Nearest Correlation Matrix—A Prob-lem from Finance.” IMA Journal of Numerical Analysis 22: 329–343.

Laurence, Peter, Tai-Ho Hwang, and Luca Barone. 2008. “Geometric Properties ofMultivariate Correlation in de Finetti’s Approach to Insurance Theory.” Elec-tronic Journal for History of Probability and Statistics 4(2).


Plerou, Vasiliki, Parameswaran Gopikrishnan, Bernd Rosenow, Luis A. Nunes Ama-ral, Thomas Guhr, and H. Eugene Stanley. 2002. “Random Matrix Approachto Cross Correlations in Financial Data.” Physical Review E 65: 1–18.

Potters,Marc, Jean-Philippe Bouchaud, and Laurent Laloux. 2005. “Financial Appli-cations of Random Matrix Theory: Old Laces and New Pieces.” Acta PhysicaPolonica B (36): 2767–2784.

Tierens, Ingrid, and Margaret Anadu. 2004. “Does It Matter Which MethodologyYou Use toMeasure Average Correlation across Stocks?” Goldman Sachs EquityDerivatives Strategy report, April 2004.

PROBLEMS

6.1 Lower Bound for Average Correlation

Let R be a n × n correlation matrix. For any n × n positive-definite matrix Adefine �̂�A(x) =

xTAx(xTe)2 where e is the vector of 1’s and x is an arbitrary vector

which is nonorthogonal to e.

(a) Show that �̂�R(e) ≤ 1n𝜆n where λn is the top eigenvalue of R.

(b) Show that �̂�R(x) ≥ [�̂�R−1 (e)

]−1. Hint: This can be formulated as a

constrained optimization problem and solved with; for example, theLagrangian method.

(c) We want to approximate the distance d = �̂�R(e) − [�̂�R−1 (e)]−1 when Ris an equity correlation matrix with top eigenvalue λn ≫ λn–1 and thecorresponding top eigenvector vn is an all-stock portfolio close to e/n (upto a scaling factor).i. Show that d may be rewritten as d = 1

n(A −H) where A, H are

respectively the arithmetic and harmonic weighted averages of theeigenvalues of R, with weights 𝛼i = cos2(̂e, vi). Hint: Use Parseval’sidentity to show that

∑ni=1 𝛼i = 1.

ii. Argue that d ≈[1−𝛼nn−1

(1 − 𝜆n

n

)+ 𝛼n𝜆nn

]− 1

1−𝛼nn−1

∑n−1i=1

n𝜆i+𝛼n

n𝜆n

6.2 Geometric Basket Call

Consider a call option with payoff max(0,bT − k) on a geometric basket

calculated as bT =n∏i=1

(S(i)T

S(i)0

)wi

where S(i)t is the price of the underlying asset

S(i) at time t and the nonnegative basket weights (wi) sum to 1.


(a) In the Black-Scholes model with constant correlation, show that underthe risk-neutral measure bT is lognormally distributed and find the dis-tribution parameters as functions of volatilities and correlations.

(b) Find a closed-form formula for the price of the call.

6.3 Worst-Of Put Pricing

Using the Black-Scholes model with constant correlation and Monte Carlosimulations, calculate the price of a one-year at-the-money worst-of putoption (see Section 1-2.3) on Apple, Microsoft, and Google, in accordancewith the following parameters:

■ Interest rate: 1%■ Dividend rates: Apple 3%, Microsoft 2.8%, Google 0%■ Volatilities: Apple 30%, Microsoft 26%, Google 23%■ Correlations: Apple-Google: 35%, Apple-Microsoft: 30%, Google-Microsoft: 50%

6.4 Continuously Monitored Correlation

Consider the LVCC model for two assets S(1) and S(2). Define the continu-ously monitored realized correlation coefficient as:

c =∫

T

0

dS(1)t

S(1)t

dS(2)t

S(2)t√√√√√∫T

0

[dS(1)t

S(1)t

]2

× ∫T

0

[dS(2)t

S(2)t

]2.

Show that c ≤ 𝜌1,2

CHAPTER 7Correlation Trading

With the development of multiasset exotic products it became possible, andat times necessary, to trade correlation more or less directly. The first corre-lation trades were actually dispersion trades where a long or short positionon a multi-asset option is offset by a reverse position on single-asset options.Recently pure correlation trades appeared in the form of correlation swaps.

7-1 DISPERSION TRADING

The payoff of a dispersion trade is of the form:

Basket Option Payoff − 𝛽 ×∑

iWeighti × Single Option Payoffi

where 𝛽 is an arbitrary coefficient or leg ratio, which is typically deter-mined so that the trade has zero initial cost, and all other notations areself-explanatory.

The intuition behind dispersion trades is that the basket option’s legprovides exposure to volatility and correlation. To isolate the correlationexposure, it is necessary to hedge, if only approximately, the volatility expo-sure: this is precisely the purpose of the short single options’ leg.

The two most popular types of dispersion trades are vanilla dispersions,based on vanilla options (typically straddles), and variance dispersions,based on variance swaps.

7-1.1 Vanilla Dispersion Trades

The payoff formula for a vanilla dispersion trade on a selection of n stocksS(1),… , S(n) with weights w1,… , wn is given as:||||||

n∑i=1

wi

S(i)T

S(i)0− k

|||||| − 𝛽n∑i=1

wi

||||||S(i)T

S(i)0− k

||||||87


where 𝛽 is the leg ratio, T is the maturity, and k is the moneyness level(strike/spot).

The trade cost is StraddleBasket0 (k,T) − 𝛽n∑i=1

wiStraddle(i)0 (k,T), and the

leg ratio for a zero-cost trade is thus 𝛽0 = StraddleBasket0∑n

i=1wi × Straddle(i)0

. In the case

of short-term near-the-money forward straddles, we have the proxy 𝛽0 ≈𝜎∗Basket∑n

i=1wi𝜎

∗i

≈√𝜌∗ATM where 𝜎*’s are at-the-money implied volatilities and

ρ*ATM is at-the-money average implied correlation (see Problem 7.1).From a trading perspective, vanilla dispersion trades are attractive

because they tend to be liquid, cost-effective, and customizable. However, amajor disadvantage is that they need to be delta-hedged; furthermore, thedelta-hedging profit and loss (P&L) involves the gammas of n + 1 optionsand is only very loosely connected to correlation:

■ Assuming constant implied volatility and zero interest rates, the totaldelta-hedging P&L for option i (where i = Basket or 1,… , n) is:

∫T

0Γ$t,i(𝜎

2t,i − 𝜎

∗2i )dt

where Γ$t,i =

12Γt,iS2t,i is the option’s dollar gamma at time t, 𝜎t,i is the

instantaneous volatility of asset S(i) at time t, and 𝜎∗i is implied volatility.■ The total delta-hedging P&L of the dispersion trade is thus:

Dispersion P&L = ∫T

0

[Γ$t,Basket

(𝜎2t,Basket

− 𝜎∗2Basket

)−𝛽

n∑i=1

wiΓ$t,i

(𝜎2t,i − 𝜎

∗2i

)]dt

Using the proxy formula from Section 6-3.1 and rearranging terms wecan write:

Dispersion P&L ≈ ∫T

0

[Γ$t,Basket𝜌t𝜎

2t − 𝛽

n∑i=1

wiΓ$t,i𝜎

2t,i

]dt

− ∫T

0

[Γ$t,Basket𝜌∗𝜎∗2 − 𝛽

n∑i=1

wiΓ$t,i𝜎

∗2i

]dt

Because the dollar gammas are all different and keep changing, the singleoption leg will likely be a poorly efficient hedge against the volatilityexposure resulting from 𝜎t.

Correlation Trading 89

7-1.2 Variance Dispersion Trades

Variance dispersion trades appeared as a spinoff of the expansion of thevariance swapsmarket and offer amuchmore direct way to trade correlationthan vanilla dispersions. Specifically, the payoff formula for a selection of nstocks S(1),… , S(n) with weights w1,… , wn is:

𝜎2Basket

− 𝛽n∑i=1

wi𝜎2i

where 𝜎’s are realized volatilities between the start date t = 0 and the matu-rity date t=T and 𝛽 is the leg ratio. Typically the n stocks are the constituentsof an equity index (or a subset), the weights are based on market capitaliza-tion at the start date, and 𝜎Basket is replaced with the volatility of the index.In practice each realized volatility is capped at a certain level to mitigatevolatility “explosions” resulting from bankruptcies, for example.

The trade cost is simply 𝜎⋆2Basket

− 𝛽n∑i=1

wi𝜎⋆2i where 𝜎⋆’s are fair

variance swap strikes, and thus the leg ratio for a zero-cost trade is

𝛽0 =𝜎⋆2Basket∑n

i=1wi𝜎⋆2i

. Note that 𝛽0 differs slightly from the correlation proxy

formula 𝜌⋆ =𝜎⋆2Basket(∑n

i=1wi𝜎⋆i

)2 and gives rise to a new average correlation

measure:

⌢𝜌(w, 𝜎) = 𝜌⎛⎜⎜⎝w1𝜎1⋮

wn𝜎n

⎞⎟⎟⎠ ×(∑n

i=1wi𝜎i

)2

∑n

i=1wi𝜎

2i

.

Note that by Jensen’s inequality we must have ⌢𝜌 ≤ 𝜌. Figure 7.1 showsthat 𝜌⋆,⌢𝜌⋆ differ very little in practice, and that they are somewhat aboveat-the-money implied correlation 𝜌∗ATM.

By definition of ⌢𝜌 we may rewrite the payoff of a zero-cost variancedispersion trade as:

(⌢𝜌 − ⌢𝜌⋆)n∑i=1

wi𝜎2i

In other words the P&L on a zero-cost variance dispersion trade is thespread between realized and implied average correlation multiplied by theaverage realized variance of the constituent stocks. As such the trade willmake money when realized correlation exceeds implied correlation, and losemoney otherwise—a remarkable property.

From a trading perspective, variance dispersion trades are attractivebecause there is a persistent gap between implied and realized correlationas shown in Figure 7.2 on the Dow Jones EuroStoxx50 index.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Ja

n-1

-02

Au

g-1

-02

Ma

r-1

-03

Oct-

1-0

3

Ma

y-1

-04

De

c-1

-04

Ju

l-1

-05

Fe

b-1

-06

Se

p-1

-06

Ap

r-1

-07

No

v-1

-07

Ju

n-1

-08

Ja

n-1

-09

Au

g-1

-09

Ma

r-1

-10

Oct-

1-1

0

Ma

y-1

-11

De

c-1

-11

Ju

l-1

-12

Fe

b-1

-13

FIGURE 7.1 Comparison between the six-month implied correlation proxy 𝜌⋆

(bold line), the six-month variance-based implied correlation ⌢𝜌⋆ (dashed line), andat-the-money implied correlation 𝜌∗ATM (thin line) for the Dow Jones EuroStoxx50 index.Data source: OptionMetrics.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Jul-1-0

2

Feb-1

-03

Sep-1

-03

Apr-

1-0

4

Nov-1

-04

Jun-1

-05

Jan-1

-06

Aug-1

-06

Mar-

1-0

7

Oct-

1-0

7

May-1

-08

Dec-1

-08

Jul-1-0

9

Feb-1

-10

Sep-1

-10

Apr-

1-1

1

Nov-1

-11

Jun-1

-12

Jan-1

-13

FIGURE 7.2 Six-month variance-based implied correlation ⌢𝜌⋆ (bold line) andrealized correlation ⌢𝜌 six months later (thin line) for the Dow Jones EuroStoxx50 index.Data sources: OptionMetrics, Bloomberg.


FOCUS ON CROSS-SECTIONAL DISPERSION

The cross-sectional dispersion of n random variables X1,… , Xn isdefined as their average squared deviation from the mean:

D = 1n

n∑i=1

(Xi −X

)2= 1n

n∑i=1

X2i −X

2

where X = 1n

∑n

j=1Xj

When dealing with time series such as stock returns wemay compute their cross-sectional dispersion through time:

Dt =1n

n∑i=1

(Xi,t −Xt)2. It then turns out that the average cross-sectional

dispersion D over m time periods matches the payoff of an equallyweighted short variance dispersion trade with leg ratio 𝛽 = 1, asshown below:

D = 1m

m∑t=1

Dt =1n

n∑i=1

1m

m∑t=1

X2i,t

⏟⏞⏞⏟⏞⏞⏟𝜎2i

− 1m

m∑t=1

X2t

⏟⏞⏟⏞⏟𝜎2Basket

7-2 CORRELATION SWAPS

7-2.1 Payoff

A correlation swap is a forward contract on average realized correlation.Specifically the payoff formula for a selection of n stocks S(1),… , S(n) withweights w1,… , wn is: ∑

i< j

wiwj𝜌i,j∑i< j

wiwj

− Kcorrel

where 𝜌’s are pairwise correlation coefficients observed between the startdate t = 0 and the maturity date t = T, and Kcorrel is the strike level between 0and 1. We recognize the average correlation measure 𝜌(w) from Section 6-3.


The main attraction of correlation swaps is that they are pure corre-lation trades. The main disadvantage is that there is no consensus modelto price and hedge them—in particular the strike Kcorrel trading on theover-the-counter market may significantly differ from average impliedcorrelation measures.

Correlation swaps are typically offered by investment banks to sophis-ticated investors on an opportunistic basis to offload the correlation riskaccumulated by their exotic trading desks. This is because exotic derivativessold by banks tend to be long correlation (i.e., their value increases whencorrelation increases), resulting in large short correlation exposures (i.e., thebank loses money when correlation increases).

7-2.2 Pricing

To approach the pricing of correlation swaps, note that when the stocksare the constituents of an equity index and weights are based on marketcapitalizations, the average realized correlation measure 𝜌(w) is usually close

to 𝜌⎛⎜⎜⎝w1𝜎1⋮

wn𝜎n

⎞⎟⎟⎠ =𝜎2Basket

−∑n

i=1w2i 𝜎

2i(∑n

i=1wi𝜎i

)2−∑n

i=1w2i 𝜎

2i

∼𝜎2Basket(∑n

i=1wi𝜎i

)2 , which itself may

be approximatedwith⌢𝜌(w, 𝜎) = 𝜎2Basket∑n

i=1wi𝜎

2i

. This suggests that a correlation

swap may be approximately priced as a derivative of two tradable assets:basket variance and average constituent variance.

Based on this observation, Bossu (2005) proposed a simple “toy model”to price and hedge correlation swaps, which is a straightforward extensionof the Black-Scholes model in the two-asset case. One important theoreticallimitation of the toy model is that it is not entirely well-specified and allowsaverage realized correlation to exceed 1 in a small number of paths.

A modified version of the toy model where correlation is constrainedbetween –1 and 1 is derived in Chapter 9. This version is more mathe-matically satisfying but unfortunately loses the simplicity of the original toymodel.

7-2.3 Hedging

While research for the ultimate correlation swap model is still ongoing, wemay state certain interesting properties. In general, two-asset models willproduce a forward price formula ft(Xt,Yt) for correlation where Xt, Yt arethe forward prices at time t of 𝜎2

Basket,∑n

i=1wi𝜎

2i respectively (in particu-

lar X0 = 𝜎⋆2Basket,Y0 =

∑n

i=1wi𝜎⋆2i , and XT = 𝜎2

Basket,YT =

∑n

i=1wi𝜎

2i ). The

hedge ratios are then given as 𝜕f𝜕X, 𝜕f𝜕Y

as usual.


Because ft(Xt,Yt) = 𝔼t(XTYT

)under the forward-neutral measure and

because XT/YT is invariant when multiplying XT and YT by the samescalar 𝜆, we must also have ft(𝜆Xt, 𝜆Yt) = ft(Xt,Yt) and thus there exists

a unique pricing function g such that ft(Xt,Yt) = gt(XtYt

)which is the

one-dimensional reduction of the two-asset model f. In this case the hedgeratios are given as 𝜕f

𝜕X= 1

Yt

𝜕g𝜕R, 𝜕f𝜕Y

= −Xt

Y2t

𝜕g𝜕R

where R = XY. Remarkably

enough the corresponding hedge is then a zero-cost variance dispersionwith leg ratio 𝛽0(t) =

||| 𝜕f𝜕Y ∕ 𝜕f𝜕X ||| = XtYt

= Rt.

This general property shows that the pricing and hedging of correlationswaps is strongly interconnected with variance dispersion trading.

PROBLEMS

7.1

Consider a zero-cost vanilla dispersion trade.

(a) Show that 𝛽0 must be comprised between 0 and 1 under penalty of arbi-trage.

(b) Derive the proxy 𝛽0 ≈𝜎∗Basket∑n

i=1wi𝜎

∗i

for at-the-money-forward straddles.

7.2

(a) In the Black-Scholes model with zero interest rates, show that the expec-tation at time 0 of the future dollar gamma of a European option athorizon t is given as:

𝔼(12ΓtS2t

)= 1

2Γ0S

20

where Γt is the option’s gamma at time t and St is the underlying assetprice at time t.Hint: Use the Black-Scholes partial differential equation and the

Ito-Doeblin theorem to show that 𝔼t[d(12𝜎2ΓtS2t

)]= 0.

(b) Consider a vanilla dispersion trade with leg ratio 𝛽. Assume zero ratesand dividends and that all realized volatilities are constant. Show thatthe expectation of the total delta-hedging P&L is then:

Γ$0,Basket

(𝜎2Basket

− 𝜎∗2Basket

)T − 𝛽n∑i=1

wiΓ$0,i(𝜎

2i − 𝜎

∗2i )T


7.3

Consider a variance dispersion trade with leg ratio 𝛽. Show that if

𝛽 =𝜎⋆

Basket∑n

i=1wi𝜎⋆

i

the portfolio is initially vega-neutral (i.e., algebraically

insensitive to changes in implied volatility).

7.4

Download the market data for the DAX and its 30 constituents fromwww.wiley.com/go/bossu and calculate the payoffs of:

■ A three-month zero-cost at-the-money vanilla dispersion trade:

||||| IndexTIndex0− 1

||||| − 𝛽030∑i=1

wi

||||||S(i)T

S(i)0− 1

|||||| where 𝛽0 =𝜎ATM∗Index∑30

i=1wi𝜎

ATM∗i

■ A three-month zero-cost variance dispersion trade

𝜎2Index

− 𝛽0n∑i=1

wi𝜎2i where 𝛽0 =

𝜎⋆2Index∑n

i=1wi𝜎⋆2i

■ A three-month correlation swap:∑i< j

wiwj𝜌i,j∑i< j

wiwj

− Kcorrel

http://www.wiley.com/go/bossu

CHAPTER 8Local Correlation

Local correlation models are a recent cutting-edge development in deriva-tives modeling and extend the concept of local volatility to multiple assets.Indeed if the volatility of each asset is thought to depend on time and thespot price, then the same idea should probably apply to the correlationcoefficient between any two assets. However there are theoretical and prac-tical issues: on the theoretical side the entire correlation matrix must remainpositive-definite, which can be challenging; and on the practical side thereare very few observable basket option prices to calibrate to. In this chapterwe give evidence of non-constant implied correlation and introduce a modelthat is consistent with this behavior.

8-1 THE IMPLIED CORRELATION SMILE ANDITS CONSEQUENCES

Just as there is an implied volatility smile for options on single stocks, there isalso an implied volatility smile for basket options, which is best observed onindex options. Figure 8.1 compares the six-month smile on the EuroStoxx 50index versus the average six-month smile on its constituents. We can see thatthe slope of the index smile is different from the slope of the constituents’smile, a phenomenon that may only be reproduced by having a differentcorrelation parameter for each moneyness level.

Specifically, for fixed maturity T, we may extract the implied correlationcurve by means of the formula:

𝜌∗(k) =𝜎∗2Basket

(k) −∑n

i=1w2i 𝜎

∗2i (k)(∑n

i=1wi𝜎

∗i

(k))2

−∑n

i=1w2i 𝜎

∗2i (k)

∼⎡⎢⎢⎣𝜎∗Basket

(k)∑n

i=1wi𝜎

∗i (k)

⎤⎥⎥⎦2

where 𝜎*’s are implied volatilities, w’s are basket weights, and k is the mon-eyness level.

95


0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

30% 65% 100% 135% 170%

FIGURE 8.1 Six-month smile on the Dow Jones EuroStoxx 50 index versusthe average six-month smile on its 50 constituents as of April 30, 2013.Data source: OptionMetrics.

Figure 8.2 shows the implied correlation smile obtained on the DowJones EuroStoxx 50 index. We can see that the shape is downward sloping,which is consistent with the intuition that whenmarkets go down correlationgoes up.

This phenomenon suggests that the constant correlation assumptionoften used to price basket exotics is not correct, opening the way for yetmore sophisticated models. This particularly affects the pricing of worst-ofand best-of options, which are very sensitive to the level of input correlation.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

30% 65% 100% 135% 170%

FIGURE 8.2 Six-month implied correlation on the Dow Jones EuroStoxx 50 indexas of April 30, 2013.

Local Correlation 97

8-2 LOCAL VOLATILITY WITH LOCAL CORRELATION

A local volatility model with local correlation (LVLC) allows the pairwisecorrelation coefficients to depend on time and spot prices:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

dS(1)t = 𝜇1S(1)t dt + 𝜎loc1

(t, S(1)t

)S(1)t dW(1)

t

dS(2)t = 𝜇2S(2)t dt + 𝜎loc2

(t, S(2)t

)S(2)t dW(2)

t

⋮

dS(n)t = 𝜇nS(n)t dt + 𝜎locn

(t, S(n)t

)S(n)t dW(n)

t

and

for all i ≠ j∶ dW(i)t dW

(j)t = 𝜌loci,j

(t, S(i)t , S

(j)t

)dt

where 𝜎loci (t, S) is the local volatility function for asset S(i) (see Chapter 4)and 𝜌loci,j (t, S

(i), S(j)) is a local correlation function for assets S(i) and S(j).There are many ways to specify the n(n–1)/2 local correlation func-

tions 𝜌loci,j and in fact some authors let them depend on the entire vector

of spot prices (S(1), S(2),… , S(n)). The key practical difficulty here is to ensurethat the local correlation matrices R(t, S(1),… , S(n)) = [𝜌loci,j (t, S

(i), S(j))]1≤i,j≤nare positive-definite at all times and across all spot levels.

It is worth nothing that the arbitrage argument leading to a pricingequation for basket options still holds in the case of LVLC models.

When delta-hedging using an LVLC model, the mismatch betweenthe option payoff and the proceeds of the delta-hedging strategy involvesn(n + 1)/2 terms corresponding to the gammas and cross-gammas. In thetwo-asset case with zero interest rates the P&L expression is:

P&L = ∫T

0Γ$1

(t, S(1)t , S

(2)t

) ⎡⎢⎢⎣(dS(1)t

S(1)t

)2

−(𝜎loc1

(t, S(1)t

))2dt

⎤⎥⎥⎦+ ∫

T

0Γ$2

(t, S(1)t , S

(2)t

) ⎡⎢⎢⎣(dS(2)t

S(2)t

)2

−(𝜎loc2

(t, S(2)t

))2dt

⎤⎥⎥⎦+ ∫

T

0Γ$1,2

(t, S(1)t , S

(2)t

)×

[dS(1)t

S(1)t

dS(2)t

S(2)t

− 𝜎loc1

(t, S(1)t

)𝜎loc2

(t, S(2)t

)𝜌loc1,2

(t, S(1)t , S

(2)t

)dt

]


where Γ$i =

12𝜕2f𝜕S(i)2

S(i)2 and Γ$1,2 = 𝜕2f

𝜕S(1)S(2)S(1)S(2). We can see that correlation

(actually covariance) only appears in the cross-gamma term, and thus theonly way to eliminate correlation exposure is by dynamically trading anotherbasket option.

In practice, exotic option traders tend to mitigate their risk by gamma-hedging their exotic position using single-stock listed vanilla options, butthey typically do not cross-gamma-hedge with basket options because theseare too illiquid and expensive.

FOCUS ON SPREAD OPTION HEDGING

The case of the spread option with payoff max(0,

S(1)T

S(1)0

−S(2)T

S(2)0

− k)

is

very informative. If correlation is very high and volatilities are similar,the dollar gamma terms tend to cancel each other and we are mostlyleft with the cross-gamma term:

P&L ≈ ∫T

0Γ$1,2(t, S

(1)t , S

(2)t )

×

[dS(1)t

S(1)t

dS(2)t

S(2)t

− 𝜎loc1 (t, S(1)t )𝜎loc2 (t, S(2)t )𝜌loc1,2(t, S(1)t , S

(2)t )dt

]

To make matters worse, the deltas also tend to cancel each other.Thus any traditional hedging strategy (delta- and gamma-hedging) israther inefficient.

This configuration may be apprehended geometrically: if x and yare two nearly collinear vectors of similar lengths, then x – y is nearlyorthogonal, and neither x nor y can efficiently be used to reproducethe spread.

x

yx – y


8-3 DYNAMIC LOCAL CORRELATION MODELS

Langnau (2010) and Reghai (2010) both explored a class of LVLC mod-els where local correlations fluctuate between two given “up” and “down”correlation matrices U and D through the convex combination1:

R(t,−→S ) = [𝜌loci,j

(t,−→S)]1≤i,j≤n = (1 − 𝛼(t,

−→S ))D + 𝛼(t,

−→S )U (8.1)

where 0 ≤ 𝛼 ≤ 1. It is easy to verify that ifD and U are positive-definite thenso is R.

Langnau shows that if the basket local volatility function 𝜎locBasket

(t,B)is known, then 𝛼 is uniquely determined. Specifically, Langnau argues that,subject to certain arbitrage conditions, we must have:[

𝜎locBasket

(t,Bt

)]2B2t =

∑i,j

wiwjS(i)t S

(j)t 𝜎

loci (t, S(i)t )𝜎locj (t, S(j)t )𝜌loci,j (t,

−→S )

where Bt =∑n

i=1wiS

(i)t is the basket price. Substituting Equation (8.1) and

solving for 𝛼 we obtain:

𝛼(t,−→S ) =

[𝜎locBasket

(t,Bt

)]2B2t − covD(t,

−→S )

covU(t,−→S ) − covD(t,

−→S )

where covA(t,−→S ) =

∑1≤i,j≤n wiwjS

(i)t S

(j)t 𝜎

loci (t, S(i)t )𝜎locj (t, S(j)t )ai,j for any

matrix A = (ai,j). Note that covA = xTAx for the vector x with entriesxi = wiS

(i)t 𝜎

loci (t, S(i)t ).

Langnau’s dynamic local correlation model reproduces the index smilevery accurately as shown in Figure 8.3 and as such constitutes a significantadvance in basket option pricing. In particular, it correctly prices basket vari-ance swaps (without producing a correct hedging strategy).

8-4 LIMITATIONS

The LVLC approach will satisfactorily price a wide range of basket exoticsand yield more accurate hedge ratios than the traditional LVCC approach.

1We slightly redefined 𝛼 versus Langnau’s paper so that R = D for 𝛼 = 0 and R = Ufor 𝛼 = 1.


0%

5%

10%

15%

20%

25%

30%

35%

40%

50% 70% 90% 110% 130% 150%

Market-implied volatility Model-implied volatility

FIGURE 8.3 The dynamic local correlation model reproduces the one-year DowJones EuroStoxx 50 index smile as of April 30, 2013, very accurately.Data source: OptionMetrics.

However, it will not generate realistic implied volatility and correlation smiledynamics for basket cliquet options, for example, for which a stochasticvolatility and correlation model would be required.

Additionally the LVLC approach will not produce a meaningful hedgingstrategy for new generation payoffs such as correlation swaps.

REFERENCES

Langnau, Alex. 2010. “A Dynamic Model for Correlation.” Risk Magazine (April):74–78.

Reghai, Adil. 2010. “Breaking Correlation Breaks.” Risk Magazine (October):90–95.

PROBLEMS

8.1 Implied Correlation

Consider a stock index made of n constituent stocks with fixed weightsw1,… , wn. For fixed maturity T, let a(k) denote the implied volatility ofthe index at moneyness k, and b(k) denote the average implied volatility ofthe constituent stocks at moneyness k. Show that implied correlation a2(k)

b2(k) isconstant if and only if the percentage slopes of a and b are equal (i.e., a′/a =b′/b).


8.2 Dynamic Local Correlation I

Consider Langnau’s dynamic local correlation model with D = I and U =eeT, where I is the identity matrix and e is the vector of 1’s. Show that

𝛼 =(𝜎locBasket

)2−∑ni=1 x

2i 𝜎

loc2i(∑n

i=1 xi𝜎loci

)2−∑ni=1 x

2i 𝜎

loc2i

for a particular choice of nonnegative weights

x summing to 1, and then argue that 𝛼 = 𝜌(x) where 𝜌(x) is defined inEquation (6.1).

8.3 Dynamic Local Correlation II

This is a continuation of Problem 8.2.Assume for ease of implementation that:

■ The Dow Jones EuroStoxx 50 index is made of 50 equally weightedconstituent stocks

■ Interest and dividend rates are zero■ All implied volatility surfaces have the following parametric form

𝜎∗(k,T) =

√𝜃2

[1 + 𝜌𝜑𝜆 (𝜃) lnk +

√(𝜑𝜆(𝜃) lnk + 𝜌)2 + 1 − 𝜌2

]where k = K/S is moneyness, 𝜃 > 0, −1 ≤ 𝜌 ≤ 1, 𝜆 ≥ 0.5 aretime-independent parameters and:

𝜑𝜆(𝜃) =1𝜆𝜃

(1 − 1 − e−𝜆𝜃

𝜆𝜃

)Download the parameters for the index and its constituents from www

.wiley.com/go/bossu and then implement Langnau’s dynamic local correla-tion model withD = I andU = eeT to price one-year arbitrary basket payoffsusing 252 time steps. Reproduce Figure 8.3 and then verify that the price ofa 50% worst-of call on the 50 constituents is approximately 11.3%.

Hint: You will need to compute time-dependent local volatilities usingEquation (4.2).



CHAPTER 9Stochastic Correlation

Stochastic correlation models may provide a more realistic approach to thepricing and hedging of certain types of exotic derivatives, such as worst-ofand best-of options and correlation swaps and correlation options. In thischapter, we review various types of stochastic correlation models and pro-pose a framework for the pricing of realized correlation derivatives that isconsistent with variance swap markets.

9-1 STOCHASTIC SINGLE CORRELATION

Consider the following general model framework for two assets S(1) and S(2):

⎧⎪⎨⎪⎩dS(1)t ∕S(1)t = 𝜇1(t,…)dt + 𝜎1(t,…)dW(1)

t

dS(2)t ∕S(2)t = 𝜇2(t,…)dt + 𝜎2(t,…)dW(2)t

(dW(1)t )(dW(2)

t ) = 𝜌(t,…)dt

where 𝜇’s are instant drift coefficients, 𝜎’s are instant volatility coefficients,and 𝜌 is the instant correlation coefficient between the driving BrownianmotionsW’s. Here all the coefficients may be stochastic, and we focus on 𝜌.

There are some simple ways tomake 𝜌 stochastic and comprised between−1 and 1; for example, take 𝜌t = sin(𝛼 + 𝛽Zt) where Z is an independentBrownian motion. The dynamics of d𝜌t may then be found by means of theIto-Doeblin theorem. One issue with this approach is that the parametersmay not be very intuitive.

A better approach is to specify diffusion dynamics for 𝜌 and examinethe Feller conditions at bounds −1 and 1 (see Section 2-4.2.2). A popularprocess here is the affine Jacobi process, also known as a Fischer-Wright

103


Trading days

0–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

50 100 150 200 250

FIGURE 9.1 Sample path of an affine Jacobi process with parameters 𝜌0 = 0.65,𝜌 = −0.1, 𝜅 = 10.6, 𝛼 = 1.

process, which is very similar to Heston’s stochastic volatility process(see Section 2-4.2.2):

d𝜌t = 𝜅(𝜌 − 𝜌t)dt + 𝛼√

1 − 𝜌2t dZt

where 𝜌 is the long-term mean, 𝜅 is the mean reversion speed, and 𝛼 isthe volatility of instant correlation. The Feller condition is then 𝛼

2

𝜅− 1 <

𝜌 < 1 − 𝛼2

𝜅. A technical analysis of this type of process can be found in van

Emmerich (2006).Figure 9.1 shows the path obtained for an affine Jacobi process with

parameters 𝜌0 = 0.65, 𝜌 = −0.1, 𝜅 = 10.6, 𝛼 = 1. Observe how all values arecomprised between −1 and 1.

9-2 STOCHASTIC AVERAGE CORRELATION

We now shift our focus to average correlation measures 𝜌(x) =

∑i<j

xixj𝜌i,j∑i<j

xixjas

introduced in Section 6-3. Because the correlation matrix R = (𝜌i,j)1≤i,j≤nmust be positive-definite at all times we cannot naively extend the singlecorrelation case with, for instance, n(n − 1)/2 affine Jacobi processes andtake their average. Note that as a consequence of positive-definiteness 𝜌(x)is actually comprised between 0 and 1 for large n.

Before we go into further detail we must distinguish between non-tradable correlation, such as rolling historical or implied correlations, and

Stochastic Correlation 105

tradable correlation, such as the historical correlation observed over a fixedtime period [0, T]:

■ Nontradable average correlation can be modeled quite freely, using, forexample, a standard Jacobi process between 0 and 1 or econometric pro-cesses such as Constant and Dynamic Conditional Correlation models(see, e.g., Engle (2009)).

■ Tradable average correlation requires special consideration to be con-sistent with other related securities such as variance swaps.

The rest of this section is devoted to the study of tradable averagecorrelation.

9-2.1 Tradable Average Correlation

Consider ⌢𝜌 = 𝜎2Basket∑ni=1 wi𝜎

2i

which was introduced in Section 7-1.2 and is related

to the proxy formula 𝜌 =(𝜎Basket∑ni=1 wi𝜎i

)2

introduced in Section 6-3.1. Because⌢𝜌 is the ratio of two tradable assets—namely, basket variance and averageconstituent variance—we can derive its dynamics from those of the twotradable assets. For example, suppose we have:

⎧⎪⎨⎪⎩dXt∕Xt = ft

(Xt,Yt

)dWt

dYt∕Yt = gt(Xt,Yt)dZt

(dWt)(dZt) = ht(Xt,Yt)dt

where Xt is the price of basket variance at time t, Yt ≥ Xt is the price ofaverage constituent variance at time t, and the driving Brownian motionsW, Z are taken under the forward-neutral measure.

Using the Ito-Doeblin theorem the resulting dynamics for ⌢𝜌 = XY

arethen:

d⌢𝜌 t∕⌢𝜌 t = (g2t − ftgtht)dt +√f 2t − 2ftgtht + g2t dBt (9.1)

where B is another standard Brownian motion constructed from W and Z.Note that, as the ratio of two prices, ⌢𝜌 t is not the price of correlation at

time t, which is why the drift coefficient in Equation (9.1) is nonzero underthe forward-neutral measure:

⌢𝜌 t = Xt

Yt=

𝔼t(XT)𝔼t(YT)

≠ 𝔼t(XT

YT

)= 𝔼t(

⌢𝜌T)


Because ⌢𝜌 is invariant when multiplying X and Y by the same scalar𝜆, we may further focus on one-dimensional reductions of the model (seeSection 7-2.3) and assume that f, g, h are functions of X/Y:

⎧⎪⎪⎪⎨⎪⎪⎪⎩

dXt∕Xt = ft

(Xt

Yt

)dWt

dYt∕Yt = gt

(Xt

Yt

)dZt

(dWt)(dZt) = ht

(Xt

Yt

)dt

In this case Equation (9.1) becomes one-dimensional; that is, the driftand volatility coefficients depend only on time and ⌢𝜌 t. This makes the fol-lowing Feller analysis considerably easier.

Omitting the time subscript for ease of exposure and using x to denotethe state variable we may rewrite Equation (9.1) as:

dx =[g2 (x) − f (x)g(x)h(x)

]xdt + x

√f 2(x) − 2f (x)g(x)h(x) + g2(x)dB

(9.2)The Feller conditions at bounds 0 and 1 are then:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

limx↓0 ∫

x0

xs (y) dy = ∞

limx↑1 ∫

x

x0

s(y)dy = ∞

s(y) = exp

(−∫

y

y0

2[g2 (u) − f (u)g(u)h(u)

]u[f 2(u) − 2f (u)g(u)h(u) + g2(u)]

du

)

Dividing both the numerator and denominator by g2(u), the integrand

in s(y) may be rewritten as 1u

[1 + 1−p2(u)

p2(u)−2p(u)h(u)+1

]with p(u) = f (u)

g(u) . Further-more,

■ As x → 0 a sufficient condition is that lim0

1−p2

p2−2ph+1 = 𝓁 ≥ 0 in which

case we have s(y) ≈ exp(−∫ y

y01+𝓁udu

)=

(y0y

)1+𝓁for y0 and y close to

0, and thus limx↓0

∫ x0x s(y)dy = ∞. A formal proof of sufficiency is proposed

in Appendix 9.A.■ As x → 1 a necessary condition is that s(y) → ∞, which in turn implies

that 1−p2

p2−2ph+1 diverges (see Appendix 9.B for a formal proof). An analysis

of this quantity over the domain p ≥ 0 and | h | ≤ 1 reveals that the onlysingularity is at (1, 1). Thus, as a corollary we have the weak necessary


condition f (u) ∼ g(u) and h(u) → 1 as u → 1. This configuration intu-itively makes sense: if average correlation is close to 1, there is almostno diversification effect, and basket variance and average constituentvariance become almost identical.

Additionally, we want f≥ g because basket variance is more volatile thanaverage constituent variance, which unfortunately makes the sufficient con-dition stated above ineffective, since p ≥ 1. We must keep all these propertiesin mind when researching suitable functions f, g, and h.

9-2.2 The B-O Model

The following model, which we call the B-O model (for beta-omega), is afurther step towards a suitable stochastic average correlation model:⎧⎪⎪⎪⎨⎪⎪⎪⎩

dXt∕Xt = 2T − tT

[𝜔 + 𝛽

(1 −

Xt

Yt

)]dWt

dYt∕Yt = 2𝜔T − tT

dZt

(dWt)(dZt) =[Xt

Yt+ 𝜔𝜔 + 𝛽

(1 −

Xt

Yt

)]dt

(9.3)

where 𝜔 is the instant volatility of constituent volatility and 𝛽 is the “ad-ditional” volatility of basket volatility.1 The corresponding dynamics forthe average correlation ⌢𝜌 ≡ x are then given by Equation (9.2) using thefunctions: ⎧⎪⎪⎨⎪⎪⎩

ft (x) = 2T − tT

[𝜔 + 𝛽(1 − x)]

gt(x) = 2𝜔T − tT

ht(x) = x + 𝜔𝜔 + 𝛽

(1 − x)

Unfortunately, both lower and upper bounds [0,1] turn out to be attract-ing in the B-O model, making it unsuitable for extreme starting values 𝜌0and long-term horizons T. However, empirical simulations exhibit plausiblepaths. Further research is needed here.

Figure 9.2 shows 10 sample paths obtained with parameters 𝜔 = 70%,𝛽 = 40% and ⌢𝜌0 = 0.5. Remarkably enough, using Monte Carlo simula-tions the price of correlation 𝔼(⌢𝜌T) in this model appears to be close to the

initial value ⌢𝜌0 = X0Y0

=𝜎⋆2Basket∑n

i=1 wi𝜎⋆2i

, also known as variance-implied correla-

tion. This suggests that the fair strike of a correlation swap on ⌢𝜌T should

1Note that dXt∕Xt = 2(𝜔 + 𝛽) T−tTdWt when ⌢𝜌 t = Xt∕Yt is equal to 0, and that

dXt∕Xt = 2𝜔T−tTdWt when ⌢𝜌 t is equal to 1.


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 9.2 Ten sample paths using the B-O model with parameters 𝜔 = 70%,𝛽 = 40%, and ⌢𝜌0 = 0.5.

be close to ⌢𝜌0, and by extension a similar result should apply to standardcorrelation swaps.

9-3 STOCHASTIC CORRELATION MATRIX

A yet more ambitious endeavor is to devise a model for the evolution ofthe entire correlation matrix Rt = (𝜌i,j(t))1≤i,j≤n through time. As pointedout earlier, the difficulty here is to ensure that Rt is positive-definite atall times.

It is worth emphasizing that, when correlations are tradable, we shouldalso ensure that the induced dynamics of average correlation ⌢𝜌 t be consistentwith variance swaps under the forward-neutral measure.

As already pointed out in Section 6-2, equity correlation matrices havestructure—namely, there is typically one large eigenvalue dominating allothers, and the associated eigenvector corresponds to an all-stock portfo-lio. As such an equity correlation matrix cannot be viewed as any kind ofrandom matrix.


Here we need to be more specific about the meaning of a (symmet-ric) random matrix. This concept was first introduced by Wishart (1928)in the form M = XXT where X is an n × n matrix of independent and iden-tically distributed random variables; the special case where X is Gaussiandeserves particular attention since it tends to the identity matrix as n → ∞.Another approach isWigner’s, wherebyM = 1

2(X +XT); a remarkable prop-

erty is that the empirical distribution of ordered eigenvalues then follows thesemi-circle law:

1n#{i∶ 𝜆i ≤ 𝜆}−−−−−→n→∞

12𝜋∫

𝜆

−2

√4 − x2dx (|𝜆| ≤ 2)

9-3.1 Spectral Decomposition and the CommonFactor Model

The empirical analysis of equity correlation matrices suggests that they maybe viewed as the sum of a (truly) random matrix and an orthogonal projec-tor onto the maximal eigenvector. Following the spectral theorem we mayindeed write:

R =

(n−1∑i=1𝜆iviv

Ti

)+ 𝜆nvnvTn

where (v1,… , vn) is an orthonormal basis of eigenvectors with eigenval-

ues 𝜆1 ≤ · · · ≤ 𝜆n. The residual matrixn−1∑i=1𝜆iviv

Ti = R − 𝜆nvnvTn may then be

approximated by a Wishart-type matrix.

For large n we could ignoren−1∑i=1𝜆iviv

Ti altogether and write:

R ≈ R̂ = (I − 𝜆nD) + 𝜆nvnvTn =⎛⎜⎜⎜⎝

1 𝜆na1a2 · · · 𝜆na1an𝜆na2a1 1 𝜆na2an

⋮ ⋱ ⋮𝜆nana1 𝜆nana2 · · · 1

⎞⎟⎟⎟⎠where a1,… , an are the entries of the maximal eigenvector vn and D =diag(a21, … , a

2n). Note that R̂ has different eigenelements fromR; however, λn

is related to average correlation because 𝜌(R̂; vn) =vTn R̂vn−1(vTn e)2−1

= 𝜆nn

1−∑ni=1 a

4i

1n(vTn e)2−

1n

∼𝜆n∕n

cos2(v̂n,e)as n → ∞.


This approach corroborates Boortz’s Common Factor Model (2008)whereby:

Rt =⎛⎜⎜⎜⎝

1 𝜉t,1𝜉t,2 · · · 𝜉t,1𝜉t,n𝜉t,2𝜉t,1 1 𝜉t,2𝜉t,n

⋮ ⋱ ⋮𝜉t,n𝜉t,1 𝜉t,n𝜉t,2 · · · 1

⎞⎟⎟⎟⎠where (ξt,1,… , ξt,n) is a vector of correlated stochastic processes in (−1, 1),such as affine Jacobi processes. One issue with the Common Factor Modelis that the (equally weighted) average realized correlation has a risk-neutraldrift, which has no particular reason to fit in the framework of Section 9-2.1.In other words the Common Factor Model does not appear to be consistentwith variance swap markets.

9-3.2 The n × n Fischer-Wright Model

Recent work by Ahdida and Alfonsi (2012) alternatively proposes the fol-lowing stochastic process for the correlation matrix Rt, which is a general-ization of the Jacobi process:

dRt =[𝜅(R − Rt

)+ (R − Rt)𝜅

]dt

+n∑i=1𝛼i

(√Rt − RtEi,iRtdWtEi,i + Ei,idW

Tt

√Rt − RtEi,iRt

)where the matrix R is the long-term correlation mean, 𝜅 = diag(𝜅1, … , 𝜅n)is a diagonal matrix of mean-reversion speeds, 𝛼 = diag(𝛼1,… , 𝛼n) is adiagonal matrix of volatility coefficients, Ei,i = diag(0,… ,0,1,0,… ,0) is thediagonal matrix with coefficient 1 at position (i,i) and 0 elsewhere,

√H

denotes the unique square root of a positive-semidefinite matrixH, and (Wt)is an n × n matrix of independent standard Brownian motions.

Subject to the condition 𝜅R+R𝜅 − (n − 2)𝛼2 being positive-semidefinite,the Ahdida-Alfonsi process is guaranteed to remain a valid correlationmatrix through time; however, a corrected Euler scheme is required forsimulation.

Unfortunately, Ahdida and Alfonsi have not studied the eigenelementsof their respective correlation matrix processes. and it is difficult to tell howrealistic their model is within the realm of equity correlation matrices. Inparticular, there is no guarantee that the induced dynamics of average corre-lation can be made consistent with realistic dynamics of basket variance and


average constituent variance in the fashion described early in the chapter.Further research is thus needed.

REFERENCES

Ahdida, Abdelkoddousse, and Aurélien Alfonsi. 2012. “A Mean-Reverting SDE onCorrelation Matrices.” arXiv:1108.5264.

Boortz, C. Kaya. 2008. “Modelling Correlation Risk.” Diplomarbeit preprint, Insti-tut fürMathematik, Technische Universität Berlin&Quantitative Products Lab-oratory, Deutsche Bank AG.

Engle, Robert. 2009. Anticipating Correlations: A New Paradigm for Risk Manage-ment. Princeton, NJ: Princeton University Press.

van Emmerich, Cathrin. 2006. “Modelling Correlation as a Stochastic Process.” Ber-gische Universität Wuppertal. Preprint.

Wishart, John. 1928. “The Generalised Product Moment Distribution in Samplesfrom a Normal Multivariate Population.” Biometrika 20A (1–2): 32–52.

PROBLEMS

9.1

Consider a stock S, which does not pay dividends, with dollar price S$, andlet X be the exchange rate of one dollar into euros. Assume that S$ andX both follow geometric Brownian motions under the dollar risk-neutralmeasure with joint dynamics:{

dS$t ∕S$t = r$dt + 𝜎dWt

dXt∕Xt = 𝜈dt + 𝜂dZt

where r$ is the constant dollar interest rate, 𝜎, 𝜈 and 𝜂 are free constantparameters, and W, Z are standard Brownian motions with stochastic cor-relation (dWt)(dZt) ≡ 𝜌tdt.(a) Show that the forward price of S quanto euro for maturity T is

S$0𝔼[exp

(r$T − 𝜎𝜂∫ T

0 𝜌tdt)]

.

(b) Assume that S0$ = $100, r$ = 0, 𝜎 = 25%, 𝜂 = 10%, d𝜌t = 𝜅(𝜌 − 𝜌t)dt +

𝛼√

1 − 𝜌2t dBt with 𝜌0 = −0.65, 𝜌 = −0.2, 𝜅 = 10.6, 𝛼 = 1. Compute theone-year forward price of S quanto euro using Monte Carlo simulationsover 252 trading days. Answer: €100.60


9.2

Consider the model for stochastic average correlation:

d⌢𝜌 t∕⌢𝜌 t = 𝜔2(1 − ⌢𝜌 t)dt + 𝜔 1 − ⌢𝜌 t√1 − ⌢𝜌 t∕2

dBt

(a) Verify that the process remains within (0,1) and that the lower bound isnon-attracting.

(b) Define h(x) =√x(2 − x). Find f(x), g(x) such that ⌢𝜌 ≡ x satisfies

Equation (9.2). Hint: Show that 1−php2−2ph+1 = 1−x∕2

1−x where p = f/g and

solve for p.(c) Do you think that this model is suitable?

APPENDIX 9.A: SUFFICIENT CONDITION FOR LOWERBOUND UNATTAINABILITY

Following the notations of Section 9-2.1, suppose that lim0

1−p2

p2−2ph+1 = 𝓁 ≥ 0.

By the definition of a limit this means that for arbitrary 𝜀 > 0 there exists an𝛼 > 0 such that:

for all 0 ≤ u ≤ 𝛼, 1 − p2(u)p2(u) − 2p(u)h(u) + 1

≤ 𝓁 + 𝜀

Thus, for all 0 < u ≤ 𝛼, −1u

[1 + 1−p2(u)

p2(u)−2p(u)h(u)+1

] ≥ −1+𝓁+𝜀u

. By integra-tion over [y0, y] ⊂ [0, 𝛼] we get:

−∫y

y0

duu

[1 +

1 − p2 (u)p2(u) − 2p(u)h(u) + 1

]≥ −∫

y

y0

1 + 𝓁 + 𝜀u

du

= −(1 + 𝓁 + 𝜀) lnyy0.

Taking exponentials:

s(y) ≥(y0y

)1+𝓁+𝜀

and thus limx↓0

∫ x0x s(y)dy = ∞ since ∫

x0

0

dy

y1+𝛽diverges for any 𝛽 ≥ 0.


APPENDIX 9.B: NECESSARY CONDITION FOR UPPERBOUND UNATTAINABILITY

Suppose that 1−p2

p2−2ph+1 converges to a finite limit 𝓁. By the definition of a

limit this means that for arbitrary 𝜀 > 0 there exists an 𝛼 < 1 such that:

for all 𝛼 ≤ u ≤ 1,𝓁 − 𝜀 ≤ 1 − p2(u)p2(u) − 2p(u)h(u) + 1

≤ 𝓁 + 𝜀

Thus, for all 𝛼 ≤ u ≤ 1, 1+𝓁−𝜀u

≤ 1u

[1 + 1−p2(u)

p2(u)−2p(u)h(u)+1

] ≤ 1+𝓁+𝜀u

. By inte-gration over [y0, y] ⊂ [𝛼, 1] we get:

(1 + 𝓁 − 𝜀) lnyy0

≤ ∫y

y0

duu

[1 +

1 − p2 (u)p2(u) − 2p(u)h(u) + 1

]≤ (1 + 𝓁 + 𝜀) ln

yy0

Taking exponentials:(y0y

)1+𝓁+𝜀≤ s(y) ≤

(y0y

)1+𝓁−𝜀

and thus limx↑1

∫ xx0s(y)dy is finite since ∫ 1

x0

dyy𝛽

converges for any 𝛽, thereby con-

tradicting the requirement that limx↑1

∫ xx0s(y)dy = ∞.

APPENDIX AProbability Review

A-1 STANDARD PROBABILITY THEORY

A-1.1 Probability Space

A probability space (Ω, , ℙ) is the provision of:

■ A set of all possible outcomes 𝜔∈Ω, sometimes called states of Nature■ A 𝜎-algebra, that is, a set of measurable events A∈, which (1) con-tains ∅, (2) is stable by complementation (A ∈ ) and (3) is stable bycountable unions (∪i∈ℕAi ∈ )

■ A probability measure ℙ: → [0, 1], which (1) satisfies ℙ(Ω) = 1 and(2) is countably additive (ℙ(

⨆iAi) =

∑iℙ(Ai) where

⨆denotes disjoint

union)

A-1.2 Filtered Probability Space

A filtered probability space (Ω, , ( t), ℙ) is a probability space equippedwith a filtration ( t), which is an increasing sequence of 𝜎-algebras (for anyt ≤ t′: t ⊆ t′ ⊆ ). Informally the filtration represents “information” gar-nered through time.

A-1.3 Independence

Two events (A, B)∈2 are said to be independent whenever their joint prob-ability is the product of individual probabilities:

ℙ(A ∩ B) = ℙ(A) × ℙ(B)

115

116 APPENDIX A: PROBABILITY REVIEW

A-2 RANDOM VARIABLES, DISTRIBUTION,AND INDEPENDENCE

A-2.1 Random Variables

A random variable is a function X: Ω → ℝ mapping every outcome witha real number, such that the event {X ≤ x}∈ for all x∈ℝ. The notationX∈ is often used to indicate thatX satisfies the requirements for a randomvariable with respect to the 𝜎-algebra .

The cumulative distribution function of X is then FX(x) = ℙ(X ≤ x),which is always defined. In most practical applications the probability massfunction ℙ(X = x) or density function fX = 𝜕FX

𝜕x(often denoted ℙ(X = x) as

well) contains all the useful information about X.The mathematical expectation of X, if it exists, is then:

■ In general: 𝔼(X) = ∫ΩX(𝜔)ℙ(d𝜔);■ For clearly discrete random variables: 𝔼(X) =

∑x∈X⟨Ω⟩ xℙ(X = x);

■ For random variables with density fX: 𝔼(X) = ∫ +∞−∞ xfX(x)dx.

The law of the unconscious statistician states that ifX has density fX theexpectation of an arbitrary function g(X) is given by the inner product of fXand g:

𝔼(g(X)) = ∫+∞

−∞g(x)fX(x)dx

if it exists.The variance of X, if it exists, is defined as 𝕍 (X) = 𝔼([X − 𝔼(X)]2) =

𝔼(X2) − [𝔼(X)]2, and its standard deviation as 𝜎(X) =√𝕍 (X).

A-2.2 Joint Distribution and Independence

Given n random variables X1,… , Xn, their joint cumulative distributionfunction is:

F(X1,… ,Xn)(x1, … ,xn) = ℙ({X1 ≤ x1} ∩ · · · ∩ {Xn ≤ xn})

and each individual cumulative distribution function FXi(xi) = ℙ(Xi ≤ xi) is

then called “marginal.”The n random variables are said to be independent whenever the joint

cumulative distribution function of any subset is equal to the product of themarginal cumulative distribution functions:

For all {i1, … , ik} ⊆ {1, … ,n}∶ F(Xi1,…Xik

) = FXi1× · · · × FXik

Appendix A: Probability Review 117

The covariance between two random variables X, Y is given as:

Cov(X,Y) = 𝔼(XY) − 𝔼(X)𝔼(Y)

and their correlation coefficient is defined as: 𝜌(X,Y) = Cov(X,Y)𝜎(X)𝜎(Y) ∈ [−1,1]. If

X, Y are independent, then their covariance and correlation is zero but theconverse is not true. If 𝜌 = ± 1 then ℙ(Y = ±aX + b) = 1.

The variance of the sum of n random variables X1,… , Xn is:

𝕍(∑n

i=1Xi

)=

∑n

i=1𝕍 (Xi) + 2

∑i<j

Cov(Xi,Xj)

If X, Y are independent with densities fX, fY, the density of their sumX + Y is given by the convolution of marginal densities:

fX+Y = fX ∗ fY∶ z → ∫+∞

−∞fX(x)fY(z − x)dx

A-3 CONDITIONING

Conditioning is a method to recalculate probabilities using known informa-tion. For example, at the French roulette the initial Ω is {0, 1,… , 36} butafter the ball falls into a colored pocket, we can eliminate several possibilitieseven as the wheel is still spinning.

The conditional probability of an event A given B is defined as:

ℙ(A|B) = ℙ(A ∩ B)ℙ(B)

Note that ℙ(A|B) = ℙ(A) if A, B are independent.This straightforwardly leads to the conditional expectation of a random

variable X given an event B:

𝔼(X|B) = ∫ΩX(𝜔)ℙ(d𝜔|B)

Generally, the conditional expectation ofX given a 𝜎-algebra ⊆ canbe defined as the random variable Y∈ such that:

For all A∈∶ ∫AX(𝜔)ℙ(d𝜔) = ∫AY(𝜔)ℙ(d𝜔)or equivalently: 𝔼(X1A) = 𝔼(Y1A). Y = 𝔼(X| ) can be shown to exist andto be unique with probability 1.

118 APPENDIX A: PROBABILITY REVIEW

The conditional expectation operator shares the usual propertiesof unconditional expectation (linearity; if X≥Y then 𝔼(X| ) ≥ 𝔼(Y| );Jensen’s inequality; etc.) and also has the following specific properties:

■ If X∈ then 𝔼(X| ) = X■ If X∈ and Y is arbitrary then 𝔼(XY| ) = X𝔼(Y| )■ Iterated expectations: if 1 ⊆2 are 𝜎-algebras then 𝔼(X|1) =𝔼(𝔼(X|2)|1). In particular 𝔼(X) = 𝔼(𝔼(X| ))

A-4 RANDOM PROCESSES AND STOCHASTICCALCULUS

A random process, or stochastic process, is a sequence (Xt) of random vari-ables. When Xt ∈ t for all t the process is said to be ( t)-adapted.

The process (Xt) is called a martingale whenever for all t < t′: Xt =𝔼(Xt′ |t).

The process (Xt) is said to be predictable whenever for all t: Xt ∈ t−(Xt is knowable prior to t).

The path of a process (Xt) in a given outcome𝜔 is the function t → Xt(𝜔).A standard Brownian motion orWiener process (Wt) is a stochastic pro-

cess with continuous paths that satisfies:

■ W0 = 0■ For all t < t′ the incrementWt′ −Wt follows a normal distribution withzero mean and standard deviation

√t′ − t

■ Any finite set of nonoverlapping increments Wt2−Wt1,Wt4

−Wt3,…

is independent.

An Ito process (Xt) is defined by the stochastic differential equation:

dXt = atdt + btdWt

whereW is a standard Brownian motion, (at) is a predictable and integrableprocess, and (bt) is a predictable and square-integrable process.

The Ito-Doeblin theorem states that a C2 function (f(Xt)) of an Ito pro-cess is also an Ito process with stochastic differential equation:

df (Xt) = f ′(Xt)dXt +12f ′′(Xt)b2t dt

APPENDIX BLinear Algebra Review

B-1 EUCLIDEAN SPACES

B-1.1 Inner Product and the Norm

A Euclidean space E is a finite-dimensional vector space for which an innerproduct ⟨⋅, ⋅⟩ is defined. The inner product, sometimes called dot productand denoted x ⋅ y, must satisfy the following three axioms:

1. Symmetry: ⟨x, y⟩ = ⟨y,x⟩2. Bilinearity: ⟨𝜆x + y, z⟩ = 𝜆⟨x, z⟩ + ⟨y, z⟩ for any 𝜆∈ℝ3. Positive-definiteness: ⟨x,x⟩ ≥ 0 and ⟨x,x⟩ = 0 if and only if x = 0

For example:

■ E = ℝn with the canonical dot product x ⋅ y =∑n

i=1 xiyi = yTx■ The space of continuous functions over the interval [a, b] with the innerproduct: ⟨f , g⟩ = ∫

b

af (x)g(x)dx

The inner product induces a distance or norm ‖x‖ =√⟨x,x⟩ with the

following standard properties:

■ Positive scalability: ‖𝜆x‖ = |𝜆|‖x‖ for any 𝜆∈ℝ■ Triangle inequality: ‖x + y‖ ≤ ‖x‖ + ‖y‖

B-1.2 Cauchy-Schwarz Inequality and Angles

The Cauchy-Schwarz inequality is one of the most important inequalities inall of mathematics and states that:

|⟨x, y⟩| ≤ ‖x‖‖y‖

119

120 APPENDIX B: LINEAR ALGEBRA REVIEW

which allows us to define the absolute angle in [0, 𝜋] between x and y as:

(̂x, y) = arccos⟨x, y⟩‖x‖‖y‖

B-1.3 Orthogonality

Two vectors are said to be orthogonal whenever their inner product is zero.A system of vectors is said to be orthogonal whenever they are pairwise

orthogonal. There can be at most n vectors in an orthogonal system, wheren is the dimension of E.

A system of vectors is said to be orthonormal when it is orthogonal andthe norm of each vector is 1.

Every Euclidean space has infinitely many orthonormal bases. In prac-tice, given an arbitrary basis (v1,… , vn), we can build an orthonormal basis(v∗1, … , v

∗n) by following Gram-Schmidt’s orthonormalization process:

⎧⎪⎪⎨⎪⎪⎩

v∗1 = v1∕ ‖‖v1‖‖u2 = v2 − ⟨v∗1, v2⟩v∗1, v∗2 = u2∕‖u2‖

⋮

un = vn −∑n−1

i=1⟨v∗i , vn⟩v∗i , v∗n = un∕‖un‖

The orthogonal projection of a given vector x onto the line Span(v) issimply ⟨x,v⟩‖v‖2 v. In ℝn with canonical inner product, the projection operator is

then P = vvT

vTv.

Given an orthonormal basis (v∗1, … , v∗n) the coordinates of a given vector

x are ⟨x, v∗i ⟩ and we have x =∑n

i=1⟨x, v∗i ⟩v∗i .An orthogonal matrix O is a square matrix whose columns and rows

are orthonormal vectors of ℝn; equivalentlyOOT =OTO = I, where I is theidentity matrix.

Parseval’s identity states that the norm of a vector does not depend onthe orthonormal basis in which its coordinates are calculated:∑n

i=1⟨x, v∗i ⟩2 = ‖x‖2

where (v∗1, … , v∗n) is an arbitrary orthonormal basis of E.

B-2 SQUARE MATRIX DECOMPOSITIONS

Given a basis = (v1,… , vn) we may represent any linear transformationf: E→ E as an n × n square matrix A whose columns are the coordinates of

Appendix B: Linear Algebra Review 121

(f (v1), … , f (vn)) in . Then F∶ ℝn → ℝn,x → Ax is an equivalent represen-tation of f.

It is often useful to rewrite a given matrix as a product or sum of simplercomponents. For example, some matrices may be written A = PDP−1 whereP is an invertible square matrix whose columns are eigenvectors and D is adiagonal matrix of eigenvalues. By the spectral theorem, this is true of everysymmetric matrix in which case P can be chosen to be orthogonal and wemay write A =

∑ni=1 𝜆iv

∗i v

∗Ti where v*’s are the columns of P. When all 𝜆’s

are positive A is said to be positive-definite and when all 𝜆’s are nonnegativeA is said to be positive-semidefinite.

Another type of decomposition is A = LU where L is lower triangularandU is upper triangular. IfA is symmetric positive-definite then we can findL, U such that U = LT and the decomposition A = LLT is called a Choleskydecomposition.

The bilinear transformation B(x, y) = yTAx defines an inner product onℝn if and only if A is symmetric positive-definite, in which case we mayrewrite:

B(x, y) =∑n

i=1𝜆i(yTv∗i )(v

∗Ti x) = ‖x‖2‖y‖2∑n

i=1𝜆i cos (̂x, v∗i ) cos (̂y, v

∗i )

where ‖x‖2 =√xTx is the canonical norm and angles are measured

canonically. The associated quantity Q(x) = xTAx = B(x,x) = ‖x‖22∑ni=1 𝜆i

cos2(̂x, v∗i ) is then known as a quadratic form.

The Rayleigh quotient ℜ(x) = xTAxxTx

= Q(x)‖x‖22

measures the scaling factor

between the canonical norm and the Q-norm; it can be shown to be com-prised between mini 𝜆i and maxi 𝜆i. The maximum eigenvalue of A is thencalled the spectral radius ρ(A), and we have ‖Ax‖22 ≤ 𝜌(A2)‖x‖22.

Solutions Manual

This Solutions Manual includes answers to all of the end of chapter prob-lems found in this book (except for a few coding problems where the

numerical answer is provided in the text).

CHAPTER 1: EXOTIC DERIVATIVES

1.1 “Free” Option

(a) The option is not really free because we may end up at a loss at andabove the strike price. (See Figure S.1.)

(b) The replicating portfolio would include selling x digital calls struck at Kat price p and buying a vanilla call struck at K for the premium of m. Inorder for the portfolio to have zero cost we must have x × p = m.

(c) The cost of one digital call using the Black-Scholes model with thegiven parameters is $0.5398. The premium of the vanilla call from theBlack-Scholes model is $7.97. Solving for x we get 7.97

0.5398= 14.76.

1.2 Autocallable

Answer: C ≈ 12%.

1.3 Geometric Asian Option

(a) From Ito-Doeblin we have ln St = ln S0 + 𝛼t + 𝜎Wt where 𝛼 = r − q −1∕2𝜎2. Substituting into the definition of AT we get:

AT = exp

(1T∫

T

0

[ln S0 + 𝛼t + 𝜎Wt

]dt

)

= exp

(1T

[T ln S0 +

T2

2𝛼 + 𝜎∫

T

0Wtdt

])which yields the required expression for AT after simplifications.

(b) From Ito-Doeblin, we get d[(T − t)Wt] = −dWt + (T − t)dWt, whichyields the required result after integration of both sides over [0, T].

123

124 SOLUTIONS MANUAL

K

FIGURE S.1 “Free” option payoff.

The distribution of ∫ T0 (T − t)dWt is thus normal with zero mean and

variance ∫ T0 (T − t)2dt = 1

3T3.

(c) Substituting ⌢𝜎 = 𝜎∕√3 and ⌢q = 12

(r + q + 𝜎

2

6

)we need to show that

ln AT is normally distributed with mean:

ln S0+(r − ⌢q − 1

2⌢𝜎2

)T = ln S0+

⎛⎜⎜⎝r − 12

(r + q + 𝜎

2

6

)− 1

2

(𝜎√3

)2⎞⎟⎟⎠T= ln S0 +

12

(r − q − 1

2𝜎2

)T

and variance 𝜎2

3T. Indeed the mean matches the expression from ques-

tion (a), and using question (b) we find that the variance of 𝜎T∫ T0 Wtdt is

𝜎2

T2 × 13T3 = 𝜎

2

3T as required.

1.4 Change of Measure

We have 𝔼ℚ(ST) = 𝔼ℙ(ST

dℚdℙ

)with dℚ

dℙ = exp(

r−𝜇𝜎WT − 1

2

(r−𝜇𝜎

)2T). Since

ST = S0e(𝜇− 1

2 𝜎2)T+𝜎WT we have after substitution:

𝔼ℚ(ST) = 𝔼ℙ{S0 exp

[(𝜇 − 1

2

(𝜎2 +

( r − 𝜇𝜎

)2))

T +(𝜎 + r − 𝜇𝜎

)WT

]}But 𝔼ℙ(exp(𝛼WT)) = e𝛼

2T∕2 and thus 𝔼ℚ(ST) = S0 exp[(𝜇 − 1

2

(𝜎2 +(

r−𝜇𝜎

)2))

T + 12

(𝜎 + r−𝜇

𝜎

)2T]. Expanding the second squared bracket and

cancelling terms we are left with 𝔼ℚ(ST) = S0erT as required.


1.6 Siegel’s Paradox

(a) Straightforward application of Ito-Doeblin.(b) The “risk-neutral” dynamics of 1/X from question (a) correspond to

the dollar risk-neutral measure, in which 1/X is non tradable (numberof euros per dollar). The paradox is resolved by introducing the eurorisk-neutral measure where 1/X follows the process:

d1Xt

= (r€ − r$)1Xt

dt + 𝜎 1Xt

dW̃t

In the dollar risk-neutral measure, 1/X is the price of the dollar-euroexchange rate quanto dollar. Following the notations of Section 1-2.4 anddefining S = 1/X, we have ρ = −1 and η = σ; thus the drift of S quanto dollarunder the dollar risk-neutral measure is r€ − r$ − ρση = r€ − r$ + σ2 asrequired.

CHAPTER 2: THE IMPLIED VOLATILITY SURFACE

2.1 No Call or Put Spread Arbitrage Condition

We know the upper bound is:

𝜕c𝜕K

=𝜕cBS𝜕K

+𝜕cBS𝜕𝜎

× 𝜕𝜎𝜕K

and we know that 𝜕c𝜕K

≤ 0 so after rearranging terms we get:

𝜕𝜎𝜕K

≤ −𝜕cBS𝜕K

∕𝜕cBS𝜕𝜎

We know the lower bound is:

𝜕p𝜕K

=𝜕pBS𝜕K

+𝜕pBS𝜕𝜎

× 𝜕𝜎𝜕K

and we know that 𝜕p𝜕K

≥ 0 so we get

𝜕𝜎𝜕K

≥ −𝜕pBS𝜕K

∕𝜕pBS𝜕𝜎

From put-call parity: c − p = S − Ke−rT , and thus:

■𝜕cBS𝜕K

= −e−rTN(d2) gives𝜕pBS𝜕K

= (1 −N(d2))e−rT

■𝜕cBS𝜕𝜎

=𝜕pBS𝜕𝜎

= e−rTK√TN′(d2)


Putting them together we get:

U − L = −

𝜕cBS𝜕K𝜕cBS𝜕𝜎

+

𝜕pBS𝜕K𝜕pBS𝜕𝜎

=e−rTN(d2)

e−rTK√TN′(d2)

+e−rT[1 −N(d2)]

e−rTK√TN′(d2)

= 1

K√TN′(d2)

Furthermore KN′(d2) = F0N′(d2) and N′(x) = 1√

2𝜋e−x

2∕2

Thus:

U − L = 1

F0√T

1√2𝜋

e−d1

2

2

=√2𝜋ed1

2

F0√T

as required.

2.2 No Butterfly Spread Arbitrage Condition

(a) Identities:■ We have:

𝜕C𝜕K

= −N(d2(K)) = ∫d2(K)

−∞e−

x2

2dx√2𝜋

Differentiating with respect to K:

𝜕2C𝜕K2

= −d2′(K)N′(d2(K)) = − d

dK

⎛⎜⎜⎜⎜⎝ln

(SK

)𝜎√T

− 𝜎2T

2𝜎√T

⎞⎟⎟⎟⎟⎠N′(d2(K))

= − ddK

(ln S

𝜎√T

− lnK

𝜎√T

− 𝜎2T

2𝜎√T

)N′(d2(K)) =

N′(d2)

K𝜎√T

as required.■ Differentiating 𝜕C

𝜕K= −N(d2(K)) with respect to 𝜎:

𝜕2C𝜕𝜎𝜕K

= 𝜕𝜕K

(K√TN′(d2))


Using the chain rule:

𝜕𝜕K

(K√TN′(d2)) =

√TN′(d2) + K

√T

ddK

{N′(d2(K))}

=√TN′(d2) + K

√Td2

′(K)N′′(d2(K))

But:

N′′(d2(K)) =−d2√2𝜋

e−d22∕2

Substituting d′2(K) and N′′(d2(K)):√

TN′(d2) + K√Td2

′(K)N′′(d2(K))

=√TN′(d2) + K

√T

−1K𝜎

√T

−d2√2𝜋

e−d22∕2

Using the fact that d2 = d1 − 𝜎√T, we get:

𝜕2C𝜕𝜎𝜕K

=√T√2𝜋

e−d2

2

2 +d2

𝜎√2𝜋

e−d2

2

2

=

(𝜎√T

𝜎√2𝜋

+d2

𝜎√2𝜋

)e−d2

2

2 =d1𝜎N′(d2)

■ Differentiating 𝜕C𝜕𝜎

= K√TN′(d2) with respect to 𝜎:

𝜕2C𝜕𝜎2

= K√TN′′(d2)d2

′(𝜎)

We have:

N′′(d2) =−d2√2𝜋

e−d22∕2

and:

d′2(𝜎) =dd𝜎

⎛⎜⎜⎜⎜⎝ln

(SK

)− 𝜎2T

𝜎√T

⎞⎟⎟⎟⎟⎠=

− ln(SK

)𝜎2

√T

−√T2


Substituting back into 𝜕2C𝜕𝜎2

= K√TN′′(d2)d2

′(𝜎):

𝜕2C𝜕𝜎2

= K√TN′(d2)

−d2√2𝜋

e−d22∕2

⎛⎜⎜⎜⎜⎝− ln

(SK

)𝜎2

√T

−√T2

⎞⎟⎟⎟⎟⎠= Kd2

√TN′(d2)

⎛⎜⎜⎜⎜⎝ln

(SK

)𝜎2

√T

+12𝜎2

√T

𝜎2√T

⎞⎟⎟⎟⎟⎠Factoring by 1

𝜎and recognizing d1 we obtain as required:

𝜕2C𝜕𝜎2

=d1d2𝜎

K√TN′(d2)

(b) First-order derivative: 𝜑′ = fx + u′fy. Second-order derivative:

𝜑′′ = (fxx + u′fxy) +[u′′fy + u′ ×

(fyx + u′fyy

)]which yields the required result after noting that fxy = fyx by Schwarz’stheorem.

(c) Applying the second-order chain rule:

𝜕2c𝜕K2

= 𝜕2C𝜕K2

+ 2𝜎∗′ 𝜕2C𝜕K𝜕𝜎

+ 𝜎∗′2 𝜕2C𝜕𝜎2

+ 𝜎∗′′ 𝜕C𝜕𝜎

Substituting the identities from (a) we get the required result after furtherstraightforward algebra.

(d) After simplifications 𝜕2c𝜕K2 ≥ 0 is equivalent to:

2d1𝜎∗′(K) + d1d2𝜎

∗′2(K) + 𝜎∗′′(K)𝜎∗(K) ≥ − 1K2T

2.3 Sticky True Delta Rule

(a) Applying the chain rule: Δ(S) = 𝜕cBS𝜕S

+ 𝜎∗′(S) 𝜕cBS𝜕𝜎

= N(d1) + 𝜎∗′(S) ×K√TN′(d2), whence the required result after substituting K = 1 and

T = 1.(b) If 𝜎∗(S) = a + bΔ(S) then 𝜎∗′(S) = bΔ′(S) and thus Δ = N(d1) +

bΔ′N′(d2) yielding the required result after appropriate substitutions.


(c) Because Δ′ > 0 (call delta goes up as S goes up) and b < 0 (volatility goesdown as S and Δ go up) the sticky true delta rule would produce a lowerdelta than Black-Scholes.

CHAPTER 3: IMPLIED DISTRIBUTIONS

3.1 Overhedging Concave Payoffs

Assume f(K1) = 0 for simplicity. From left to right: start with f ′(K1) callsstruck at K1 so as to be tangential to the payoff; add 𝛼 = f (K3)−f ′(K1)(K3−K1)

K3−K2calls struck at K2 such that the portfolio matches the payoff f(K3) at K3;then add 𝛽 = f ′(K3) − f ′(K1) − 𝛼 calls struck at K3 so as to be tangential tothe payoff after K3; and so on (see Figure S.2).

K2 K3K1

FIGURE S.2 Over-hedging concave payoffs.

3.2 Perfect Hedging with Puts and Calls

From Section 3-2:

f (ST) = f (0) + f ′(0)ST + ∫∞

0f ′′(K)max(0, ST − K)dK


Splitting the integral at F and using terminal put-call parity cT = ST −K + pT we get:

f (ST) = f (0) + f ′(0)ST + ∫F

0f ′′(K)[(ST − K) + pT]dK + ∫

∞

Ff ′′(K)cTdK

= f (0) + f ′(0)ST + ST∫F

0f ′′(K)dK − ∫

F

0f ′′(K)KdK

+ ∫F

0f ′′(K)pTdK + ∫

∞

Ff ′′(K)cTdK

But ∫ F0 f ′′(K)dK = f ′(F) − f ′(0), and:

∫F

0f ′′(K)KdK = [f ′(K)K]F0 − ∫

F

0f ′(K)dK = Ff ′(F) − f (F) + f (0)

After substitutions and simplifications we get:

f (ST) = [f (F) − f ′(F)F] + f ′(F)ST + ∫F

0f ′′(K)max(0,K − ST)dK

+ ∫∞

Ff ′′(K)max(0, ST − K)dK

Thus:

f0 = f (F)e−rT + ∫F

0f ′′(K)p0(K)dK + ∫

∞

Ff ′′(K)c0(K)dK

because 𝔼(ST) = F.

3.3 Implied Distribution and Exotic Pricing

(b) ■ Answer: ≈ .1043■ Answer: ≈ 1.0789■ Answer: ≈ .2693■ Answer: ≈ .0211. Vanilla overhedge with strikes 0.5, 0.9, 1, 1.1, 1.3,1.7: Quantities are 0, 0, 0.0100, 0.1200, 0.6700, and 1.5200 (seeFigure S.3).

(c) (i) Answer: approximately $0.8965.(ii) Answer: approximately $0.8626.


0.2

0.4

0.6

0.8

1

1.2

1.4

00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

FIGURE S.3

3.5 Path-Dependent Payoff

(a) For example: forward start option, Asian option.(b) Pseudo-code:

1. Loop for i = 1 to N:a. Generate and calculate s1 ∶= S0 exp

((r − 1

2𝜎2

)T1 + 𝜎𝜀1

√T1

)b. Generate 𝜀1 ∼ (0,1) and 𝜀2 ∼ (0,1) independently and calcu-

late

s2 = s1 exp((

r − 12𝜎2

)(T2 − T1) + 𝜎𝜀2

√T2 − T1

)c. Calculate Payoff (i) = f (s1, s2)

2. Return e−rT2N

N∑i=1

Payoff (i)

(c) We only know the marginal distributions of ST1, ST2

but we are missingthe conditional implied distribution of ST2

|ST1.


3.6 Delta

From 𝜎∗(S0,K) ≡ 𝜎∗(K∕S0) we get 𝜕𝜎∗

𝜕K≡ 1

S0𝜎∗′, 𝜕

2𝜎∗

𝜕K2 ≡ 1S20

𝜎∗′′ and thus:

ℙ{ST = K} =N′(d2)

K𝜎∗√T

[1 + 2d1

(KS0𝜎∗′

√T)+ d1d2

(KS0𝜎∗′

√T)2

+(KS0𝜎∗′′

√T)(

KS0𝜎∗

√T)]

atm_vol = SVI(1);atm_vol1 = SVI(1/(1+epsilon));

price = 0;for i=1:n

normal = randn;x = exp(atm_vol*normal*sqrt(T) - 0.5*atm_vol∧2*T);x1 = (1+epsilon)*exp(atm_vol1*normal*sqrt(T) - 0.5*atm_vol1∧2*T);price = (i-1)/i*price + Payoff(x)*ImpDist(x) ...

/ lognpdf(x,-0.5*atm_vol∧2*T,atm_vol*sqrt(T)) / i;price1 = (i-1)/i*price1 + Payoff(x1)*ImpDist(x1) ...

/ lognpdf(x1,-0.5*atm_vol1∧2*T,atm_vol1*sqrt(T)) / i;enddelta = price - price1

CHAPTER 4: LOCAL VOLATILITY AND BEYOND

4.1 From Implied to Local Volatility

(a) We rewrite Equation (4.1) by solving for 𝜕C𝜕T

after using the result from

Problem 2.2(c) for 𝜕2C𝜕K2 .

𝜎2loc(T,K) =

𝜕C𝜕T

12K2 𝜕

2C𝜕K2

𝜕2C𝜕K2

=N′(d2)

K𝜎∗(K)√T

[1 + 2d1

(K𝜕𝜎𝜕K

√T)+ d1d2

(K𝜕𝜎𝜕K

√T)2

+(K𝜕2𝜎

𝜕K2

√T)(K𝜎∗(K)

√T)

]


Take C(K,T) = CBS(S,K,T, 𝜎∗(KT)) and derive with respect to T:

𝜕C𝜕T

=𝜕CBS

𝜕T+ 𝜕𝜎

∗

𝜕T

𝜕CBS

𝜕𝜎

We are given 𝜕CBS𝜕T

and 𝜕CBS𝜕𝜎

so:

𝜕C𝜕T

=KN′(d2)𝜎∗

2√T

+ 𝜕𝜎∗

𝜕TK√TN′(d2)

We then plug in 𝜕C𝜕T

and 𝜕2C𝜕K2 into Equation (4.1):

KN′(d2)𝜎∗

2√T

+ 𝜕𝜎∗

𝜕TK√TN′(d2)

12K2

N′(d2)

K𝜎∗√T

×

[1 + 2d1

(K𝜕𝜎∗

𝜕K

√T)+ d1d2

(K𝜕𝜎∗

𝜕K

√T)2

+(K𝜕2𝜎∗

𝜕K2

√T)(

K𝜎∗√T)]−1

By dividing both numerator and denominator by KN′(d2)2𝜎∗

√T, we obtain the

desired Equation (4.2):

𝜎2loc(T,K) = 𝜎∗2

×1 + 2T𝜎∗𝜕𝜎∗

𝜕T

1 + 2d1(K𝜕𝜎∗

𝜕K

√T)+ d1d2

(K𝜕𝜎∗

𝜕K

√T)2

+(K𝜕2𝜎∗

𝜕K2

√T)(K𝜎∗

√T)

(b) Replace 𝜎2loc(T,K) with 𝜎2

loc(T) so we now have:

𝜎∗2(T)1 + 2T𝜎∗𝜕𝜎∗

𝜕T1 + 0 + 0 + 0

= 𝜎∗2(T)(1 + 2T𝜎∗𝜕𝜎∗

𝜕T

)Now using the Hint:

ddT

(T𝜎∗2(T)) = 1 × 𝜎∗2(T) + T × 2 × 𝜎∗(T)𝜕𝜎∗

𝜕T= 𝜎∗2(T) + 2T𝜎∗(T)𝜕𝜎

∗

𝜕T

= 𝜎∗2(T)(1 + 2T𝜎∗𝜕𝜎∗

𝜕T

)


Thus 𝜎2loc(T) = 𝜎∗2(T)

(1 + 2T𝜎∗𝜕𝜎∗

𝜕T

)and by integration we find that:

1T∫

T

0𝜎2loc(t, St)dt =

1T∫

T

0

ddt

(t𝜎∗2(t))dt = 𝜎∗2(T)

as required.(c) To establish the hint, note that 𝜕

2𝜎∗

𝜕K2 = 0 by linearity of 𝜎*, and substitute

d2 = d1 − 𝜎∗√T. Differentiating 𝜎loc(K) ≈

𝜎∗(K)1+d1K𝜎∗′(K)

√Twith respect to

K we obtain:

𝜎loc′(K) ≈ 𝜎∗′(K)

1 + d1K𝜎∗′(K)

√T

+ 𝜎∗(K)− ddK

(d1K)

(1 + d1K𝜎∗′(K)

√T)2𝜎∗′(K)

√T

Near the money both denominators are close to 1; furthermore

d1 =ln(S∕K)+ 1

2 𝜎∗2T

𝜎∗√T

gives ddKd1 = − 1

K𝜎∗√T− ln(S∕K)𝜎∗′

√T

𝜎∗2T+ 1

2𝜎∗′

√T ≈

− 1

K𝜎∗√T+ 1

2𝜎∗′

√T. After substitution and simplification:

𝜎loc′(K) ≈ 𝜎∗′(K) + 𝜎∗′(K)

[1 − 1

2𝜎∗′ (K)T − d1𝜎

∗′(K)√T]

whose leading term is 2𝜎∗′(K).

4.2 Market Price of Volatility Risk

(a) The delta-hedged portfolio Π is long one option and short 𝛿S =𝜕f𝜕S

unitsof S. Thus:

dΠ = df − 𝛿SdS =(𝛾f ,tdt +

𝜕f𝜕SdS +𝜕f𝜕vdv

)− 𝛿SdS

which yields the required result after cancelling terms.(b) We have:

dΠt − rΠtdt =(𝛾f,tdt +

𝜕f𝜕vdv

)− r

(ft −𝜕f𝜕SSt

)dt

=(𝛾f,t − rft + r

𝜕f𝜕SSt + 𝛼t𝜕f𝜕v

)dt + 𝜔t

𝜕f𝜕vdZt

But rft − rSt𝜕f𝜕S

− 𝛾f,t = Λt𝜕f𝜕v

which yields the required result after substi-tution and simplifications.


(c) Conditional expectation: 𝔼t(dΠtΠt

)=

[r +𝜆t𝜔t𝜕f𝜕v

Πt

]dt. Conditional stan-

dard deviation: √𝕍t

(dΠt

Πt

)=𝜔t

|||||𝜕f𝜕v||||||Πt| dt

Thus 𝔼t(dΠtΠt

)= rdt ± 𝜆t

√𝕍t

(dΠtΠt

), that is, at time t the risk-neutral

expected return on the delta-hedged portfolio is the risk-free interest rateplus a positive or negative risk premium,which is proportional to the riskof the portfolio, with proportionality coefficient λ. This result appliesto any option and λ is independent from the particular option chosen,which is why it is called the market price of volatility risk.

4.3 Local Volatility Pricing

(a) Figure S.4 shows a graph of the corresponding local volatility surface.(b) Using Monte Carlo simulations:

■ “Capped quadratic” option: min(1,

S21

S20

); answer: 0.76949

■ Asian at-the-money-call: max(0, S0.25+S0.5+S0.75+S1

4×S0− 1

); answer:

0.1242■ Barrier call: max(0, S1 − S0) if S always traded above 80 using 252daily observations, 0 otherwise. Answer: 12.87577

1

2

3

0.5

1.5

2.5

0

00 50 150100 200

5

FIGURE S.4 Local volatility surface.


CHAPTER 5: VOLATILITY DERIVATIVES

5.1 Delta-Hedging P&L Simulation

(a) Positive Path: See Figure S.5Negative Path: See Figure S.5

(b) Average P&L: −$11,663.80 (see Figure S.6)

12,000

10,000

8,000

6,000

4,000

2,000

–2,000

00 0.2 0.4 0.6 0.8 1 1.2

0 0.2 0.4 0.6 0.8 1 1.2

–45,000–40,000–35,000–30,000–25,000–20,000–15,000–10,000

–5,0000

FIGURE S.5 Cumulative P&L: positive and negative paths.

0

10,0

00

–10

,000

–20

,000

–30

,000

–40

,000

–50

,000

–60

,000

–70

,000

–80

,000

–90

,000

–10

0,00

0

–11

0,00

0

–12

0,00

0

20,0

0030

,000

40,0

0050

,000

60,0

0070

,000

80,0

0090

,000

100,

000

110,

000

120,

000

130,

000

0500

1,0001,5002,000

FIGURE S.6 Distribution of final cumulative P&L.

5.2 Volatility Trading with Options

(a) (i) If 𝜎 is constant then Cumulative P&L = 12(𝜎2 − 𝜎∗2)∫ T

0 er(T−t)ΓtS2t dt.For a vanilla call Γ > 0 and thus the cumulative P&L is always positive.


(ii)We have Cumulative P&L = 12∫ T0 er(T−t)𝔼[ΓtS2t (𝜎

2t − 𝜎∗2)]dt. By iter-

ated expectations:

𝔼[ΓtS2t

(𝜎2t − 𝜎

∗2)] = 𝔼[ΓtS2t 𝔼t

(𝜎2t − 𝜎

∗2)] > 0(b) The aggregate cumulative P&L is:

q12 ∫

T

0er(T−t)Γ(1)

t S(1)2t

[𝜎(1)2t − 𝜎∗(1)2

]dt+

q22 ∫

T

0er(T−t)Γ(2)

t S(2)2t

[𝜎(2)2t − 𝜎∗(2)2

]dt

As a sum of two independent normal distributions the aggregate distri-bution is normal with parameters:■ Mean: q1m1 + q2m2

■ Standard deviation:√q21s

21 + q22s

22 ≤ (

q1 + q2)max(s1, s2)

This result sheds light on the diversification effect obtained by delta-hedging several options within an option book: the expected incomeis the sum of individual delta-hedging P&Ls, but it is less risky thandelta-hedging a single position.

5.4 Generalized Variance Swaps

(a) From Ito-Doeblin: d[g(St)] = g′(St)dSt +12g′′(St)𝜎2t S

2t dt. But g

′′(S) = f (S)S2

;

rearranging terms we get: 12f (St)𝜎2t dt = d[g(St)] − g′(St)dSt. Integrating

over [0, T] yields the desired result.Thus generalized variance may be replicated by a combination of cash,two derivative contracts paying off g(ST) at maturity, and dynamicallytrading S to maintain a short position of 2g′(St) at all times. Takingexpectations we find that the fair value is:

𝔼

(∫

T

0f(St)𝜎2t dt

)= 2𝔼[g(ST)] − 2g(S0) = 2(g0 − g(S0))

since 𝔼(dWt) = 0.(b) From Problem 3.2 we get: g0 = g(S0) + ∫ S0

0 g′′(K)p0(K)dK + ∫ ∞S0g′′(K) ×

c0(K)dKwhence the required formula after substituting g′′(K) = f (K)K2 and

annualizing.

(c) For the corridor varswapKgvar =√

2T

(∫ S0L

1K2 p (K)dK + ∫ U

S01K2 c(K)dK

)


5.5 Call on Realized Variance

(a) Solving for vt we have: vt = v0 exp(−2𝜔2∫ T

0

(T−tT

)2dt + 2𝜔∫ T

0T−tTdWt

)Thus vt is lognormally distributed with mean −2𝜔2∫ T

0

(T−tT

)2dt =

−23𝜔2T and standard deviation 2√

3𝜔√T. From Black’s formula we have

Varcall0 = e−rT[v0N(d1) − KN(d2)] where d1,2 =ln(v0∕K)±

23𝜔

2T

2𝜔√T∕3

. At the

money this simplifies to Varcall0 = v0e−rT[N(𝜔

√T∕3) −N(−𝜔

√T∕3)]

which yields the required formula because N(x) − N(−x) = 2N(x) − 1.(b) For x ≈ 0 we have N(x) ≈ N(0) + xN′(0).

CHAPTER 6: INTRODUCING CORRELATION

6.1 Lower Bound for Average Correlation

(a) Substitute x = e in 𝜌R(x) ≤ 1n𝜆n∕cos2(̂x, e).

(b) Because 𝜌R(𝛼x) = 𝜌R(x) we have minx∈ℝn𝜌R(x) = min

x∈ℝn;xTe=1(xTRx). Define

the Lagrangian (x, 𝜆) = xTRx − 𝜆(eTx − 1). The first-order conditionyields 2xTR = −𝜆eT that is, x = 1

2𝜆R−1e. The constraint eTx = 1 then

gives the value of λ and we get the required result after substitution andsimplification.

(c) (i) By spectral decomposition: eTRe =∑n

i=1 𝜆i(vTi e)2, eTR−1e =∑n

i=1 𝜆−1i (vTi e)

2. Furthermore, Parseval’s identity states that∑ni=1 (vTi x)

2 =xTx for any x, and thus

∑ni=1 (vTi e)

2 = n. Scaling by appropriateconstants we obtain the required result.(ii) Rewrite A =

∑n−1i=1 𝛼i𝜆i + 𝛼n𝜆n and make the approximation that for

i = 1,… , n − 1:

𝛼i ≈1

n − 1

n−1∑j=1𝛼j =

1n − 1

(1 − 𝛼n)

which is justified by the fact that 𝛼n ≈ 1 since vn and e are assumed toform a tight angle. Proceed similarly for H.


6.2 Geometric Basket Call

(a) We have:

lnbT =n∑i=1

wi lnS(i)T

S(i)0=

n∑i=1

wi

[(𝜇i −

12𝜎2i

)T + 𝜎iW

(i)T

]

=n∑i=1

wi

(𝜇i −

12𝜎2i

)T +

n∑i=1

wi𝜎iW(i)T

where 𝜇i =1Tln F(i)

S(i)0

is the annualized risk-neutral drift of S(i). As a

sum of normal variables ln bT is normally distributed with mean

m =n∑i=1

wi

(𝜇i −

12𝜎2i

)T and variance:

v = 𝕍

(n∑i=1

wi𝜎iW(i)T

)=

T∑i=1

w2i 𝜎

2i T + 2

∑i<j

wiwj𝜎i𝜎j𝜌i,jT

(b) price = e−rT𝔼[max

(0, em+𝜀

√v−k

)]= e−rT+

12 v𝔼

[max

(0, em−

12 v+𝜀

√v − k′

)]where k′ = ke−

12 v. Using Black’s formula: price = e−rT+v∕2[emN(d1) −

k′N(d2)] with d1,2 =m−lnk′± 1

2 v√v

. Further simplifications are possible.

6.4 Continuously Monitored Correlation

After substitution, we have by Cauchy-Schwarz:

c =∫

T

0𝜎(1)t 𝜎

(2)t 𝜌1,2dt√

∫T

0[𝜎(1)t ]2dt × ∫

T

0[𝜎(2)t ]2dt

= 𝜌1,2⟨𝜎(1), 𝜎(2)⟩‖𝜎(1)‖ ⋅ ‖𝜎(2)‖ ≤ 𝜌1,2,

where ⟨f , g⟩ = ∫ T0 f (t)g(t)dt.


CHAPTER 7: CORRELATION TRADING

7.1

(a) 𝛽0 = StraddleBasket0∑ni=1 wi×Straddle

(i)0

is positive because straddles always have a positive

payoff and thus price. It must be less than 1 because of the triangleinequality: |||∑n

i=1wiS(i)T∕S(i)0 − k||| ≤ ∑n

i=1wi|S(i)T ∕S(i)0 − k|(b) Use the proxy StraddleATMF

0 ≈ 2√2𝜋S0𝜎

∗√T

7.2

(a) From the Black-Scholes PDE 12𝜎2ΓtS2t = −Θt. Applying Ito-Doeblin toΘ:

d[12𝜎2ΓtS2t

]= −dΘt = −

[𝜕Θt𝜕tdt + 𝜕Θt

𝜕SdSt +

12𝜕2Θt𝜕S2

(dSt

)2]. Taking expec-

tations we get:

𝔼td[12𝜎2ΓtS2t

]= −

[𝜕Θt

𝜕t+ 1

2𝜕2Θt

𝜕S2𝜎2S2t

]dt

which vanishes because Greeks must also satisfy the Black-Scholes PDE.

7.3

At time 0 the portfolio value is 𝜎⋆2Basket

− 𝛽n∑i=1

wi𝜎⋆2i , and the vega of each com-

ponent is 𝜕(𝜎⋆2)𝜕𝜎⋆

= 2𝜎⋆. Hence the portfolio vega is 2𝜎⋆Basket

− 2𝛽n∑i=1

wi𝜎⋆i =

2𝜎⋆Basket

− 2𝜎⋆

Basket∑ni=1 wi𝜎

⋆i

n∑i=1

wi𝜎⋆i = 0.

CHAPTER 8: LOCAL CORRELATION

8.1 Implied Correlation

Implied correlation is constant iff ddk

[a(k)b(k)

]2= 0, i.e., iff 2

[a(k)b(k)

]a′(k)b(k)−a(k)b′(k)

b2(k)= 0, whence the required result after simplifications.


8.2 Dynamic Local Correlation I

When D = I and U = eeT Langnau’s alpha is:

𝛼 =

(𝜎locBasket

)2B2t − yTIy

yT(eeT)y − yTIy=

(𝜎locBasket

)2B2t − yTy

(yTe)2 − yTy

where yi = wi𝜎loci (t, S(i)t )S(i)t . Define xi =

wiS(i)t

Btand divide both the numerator

and denominator of the above expression by B to get:

𝛼 =

(𝜎locBasket

)2−

∑ni=1 x

2i 𝜎

loc2i(∑n

i=1xi𝜎

loci

)2−

∑n

i=1x2i 𝜎

loc2i

= 𝜌 (x)

It is easy to verify that∑n

i=1 xi = 1.

CHAPTER 9: STOCHASTIC CORRELATION

9.2

(a) The process clearly remains within [0,1] because it is continuous and itsdrift and volatility coefficients vanish at 0 and 1. Let us show that thebound 0 is nonattracting:

s(y) = exp(−∫

y

y0

2𝜔2x (1 − x)x2𝜔2(1 − x)2∕(1 − x∕2)

dx)

= exp(−∫

y

y0

2 − xx (1 − x)

dx)

= exp(−[2 lnx − ln (1 − x)

]yy0

)=

(yy0

)−2 ( 1 − y1 − y0

)

Thus limx↓0

∫ xx0s(y)dy = ∞ since ∫ x0

0dyy2

diverges. A similar analysis shows

that the bound 1 is also nonattracting.


(b) We want to find f(x), g(x) such that:

dx∕x = (g2 − fgh)dt +√f 2 − 2fgh + g2dBt.

Thus we must solve:⎧⎪⎨⎪⎩g2 − fgh = 𝜔2 (1 − x)

f 2 − 2fgh + g2 = 𝜔2 (1 − x)2

1 − x∕2

Taking ratios we get g2−fghf2−2fgh+g2 = 1−x∕2

1−x ; dividing both numerator and

denominator by g2 on the left-hand side we obtain 1−php2−2ph+1 = 1−x∕2

1−x .

Solving for p after substituting h =√x(2 − x) we find p =

√x

2−x , that

is, f = g√

x2−x . Substituting in g2 − fgh = 𝜔2(1 − x) and simplifying we

get g2 = 𝜔2, and thus f = 𝜔√

x2−x .

(c) This model is not suitable for several reasons: it has only one parameter𝜔, which leaves little freedom for parameterization, and the volatility ofbasket variance f is lower than the volatility of constituent variance g,which is contrary to empirical observation.

Author’s Note

This is a book about finance, intended for professionals and future profes-sionals. I am not trying to sell you any security, or give you any investment

advice. The views expressed here are solely mine and do not necessarilyreflect those of any entity directly or indirectly related tome. I took great carein proofreading this book, but I disclaim any responsibility for any remainingerrors and any use to which the contents of this book is put. Some chapterscontain original research material whose accuracy cannot be guaranteed.

143

About the Author

Sébastien Bossu is currently Principal at Ogee Group LLC where he runsthe Ogee Structured Opportunities fund, which posted a 29.8 percent net

return in 2012–2013. He has almost 10 years’ experience in banking andthe financial industry at institutions such as J.P. Morgan, Dresdner Klein-wort, and Goldman Sachs. An expert in derivative securities, Sébastien haspublished several papers and textbooks in the field and is a regular speakerat international conferences.

Sébastien is currently an Adjunct Professor at Pace University and wasrecently inducted into the 2014 edition of Who’s Who in America, pub-lished by Marquis. He is a graduate from the University of Chicago, HECParis, Columbia University, and Université Pierre et Marie Curie, and hepreviously taught at Fordham University, HEC Paris, and Université Paris-7.

145

Index

AAsian options, 2, 12Attainable boundaries, 28Attracting boundaries, 28Autocallable options, 12Average correlation, 77–82, 85,

104–108

BBarrier options, 2–3Basket options, 5–6, 87–88, 95–99Basket variance, 105, 106Best-of options, 6Binomial trees, 45–46Black-Scholes model, 1–5, 6, 26,

34–35, 93with constant correlation, 82–83,

86and volatility trading, 59–60

B-O model, 107–108Boortz’s Common Factor Model,

109Breeden-Litzenberger formula, 36Brownian motion, 7–8, 46–47, 83,

117Butterfly spreads, 33–35and Dupire’s equation, 48no arbitrage condition, 22

CCalendar spreads, 22, 48Capital Guaranteed Performance

Note, 10Carr-Wu model. See LNV modelCauchy-Schwarz inequality, 80,

119–120Change of measure, 12Change of numeraire, 7–9Chicago Board Options Exchange

(CBOE), 60, 67Cholesky decomposition, 83, 121Cliquet options, 4, 99Common factor model, 109–110Conditioning, 117–118Constant correlationBlack-Scholes model with,

82–83, 86local volatility with (LVCC), 84,

86Correlation, 73–86. See also

Local correlation; Stochasticcorrelation; Dispersion trading

average correlation, 77–82, 85,104–108

Black-Scholes with constantcorrelation, 82–83, 86

continuously monitored, 86, 140

147

148 INDEX

Correlation (Continued)correlation matrices, 75–77,

85–86, 108–110correlation proxy, 77–82,

86correlation swaps, 91–93historical, 73–74, 104implied, 75, 95–96, 100,

104measuring, 73–75trading, 87–94

Cross-sectional dispersion, 91Cubic spline interpolation,

21

DDaily volatility rule, 2De Finetti formula (fifth property of

Euclidean metric), 82Delta, 44delta hedging, 59–60, 62, 70, 88,

97and implied volatility smile, 18sticky-delta rule, 18, 31

Digital options, 1–2, 18–19Dispersion trading, 87–91cross-sectional dispersion, 91vanilla dispersion trades, 87–88,

93variance dispersion trades,

89–91, 94Dollar gamma, 59–60Dupire’s equation, 46, 47–48,

56Dynamic correlation models,

98–99, 101

EEigenvalues, 75, 85, 109Equity correlation matrix, 75–76,

108, 109Equity-linked notes, 9–10Euclidean metric (fifth property of),

82Euclidean spaces, 119–120Euler-Maruyama discretization, 46,

47, 53European payoff pricing and

replication, 35–39, 39–41Exotic derivatives, 1–11, 121Asian options, 2, 12barrier options, 2–3basket options, 5–6, 87–88,

95–99cliquet options, 4, 99digital options, 1–2, 18–19forward start options, 4, 54lookback options, 3, 12, 25multi-asset options, 4quanto options, 7–9ratchet options, 4spread options, 4–5, 98structured products, 9–11worst-of/best-of options, 6

Extrapolation, 19–21

FFeller condition, 27–28, 103, 106,

107Fifth property of Euclidean metric,

82Filtered probability space,

115

Index 149

Fischer-Wright model, 103–104,110. See also Jacobi process

Forward start options, 4, 54Forward variance, 64–65

GGamma, 59–60. See also Dollar

gammaGeneralized variance swaps, 71Geometric basket call, 85–86Girsanov theorem, 7, 14Gram-Schmidt’s

orthonormalization process,120

Greeks, 41. See also Delta; GammaGurrieri’s model, 50

HHedge ratios, 15–18Hedging. See also Overhedgingcorrelation swaps, 92–93delta-hedging, 59–60, 62, 70, 88,

97, 136with local volatility, 49variance swaps, 62–64

Hedging theory (of stochasticvolatility models), 51–52

Heston model, 26–28, 51, 53, 54,103

Historical correlation, 73–74, 104Historical volatility, 59

IImplied correlation, 75, 95–96,

100, 104

Implied distributions, 33–44butterfly spreads and, 33–35European payoff pricing and

replication, 35–41and exotic pricing, 42, 131Greeks, 41and overhedging, 37–39problems, 42–44references, 42solutions, 130–133

Implied volatility, 59local volatility from, 49, 55–56market- vs. model-implied, 23

Implied volatility derivatives,67–69

VIX futures, 67–68VIX options, 68–69

Implied volatility smile, 15–19,25

Implied volatility surface, 15–32Gurrieri’s model, 50indirect models, 25interpolation and extrapolation,

19–21LNV model, 28–29, 51properties, 22SABR model, 25–26SVI model, 23–25

Independence, 115, 116–117Interpolation, 19–21Ito-Doeblin theorem, 8, 14, 51, 63,

83, 93, 103, 105, 117

JJacobi process, 103–104, 110Jensen’s inequality, 66, 89

150 INDEX

Joint distribution, 116–117Jumps, 54–55

KKnock-in/knock-out option, 3Kolmogorov equation, 56

LLaw of the unconscious statistician,

116LNV model, 28–29, 51Local correlation, 95–101dynamic models, 98–99, 101implied correlation smile, 95–96local volatility model with

(LVLC), 96–98, 99–100Local volatility, 45–49and binomial trees, 45–46calculating, 47–50connection with stochastic

volatility, 53Dupire’s equation, 46, 47–48, 56hedging with, 50from implied volatility, 49–50,

55–56, 133–135with local correlation (LVLC),

96–98, 99–100model with constant correlation

(LVCC), 84, 86pricing, 56

Log-Normal Variance (LNV)model, 28–29, 51

Lookback options, 3, 12, 125LVCC model, 84, 86LVLC model, 96–98, 99–100

MMarket capitalization weights,

77–78Market price of volatility risk, 55Market-implied volatility, 23Matrices:correlation, 75–77, 85–86,

108–110square matrix decompostion,

120–121Milstein’s discretization method, 47Model-implied volatility, 23Monte Carlo simulations, 46,

53–54, 83, 107Multi-asset options, 4

NNo butterfly spread arbitrage

condition, 30–31No call or put spread arbitrage

condition, 30Non-attracting boundaries, 28, 32,

129Nontradable average correlation,

104Normalized liquidity rule, 2

OOrthogonality, 120Overhedging, 37–39, 42, 130

PParseval’s identity, 120Path-dependent payoff, 44, 131Pearson’s correlation coefficient, 73

Index 151

Perfect hedging with puts and calls,42, 130

Probability space, 115Probability theory, 115

QQuanto options, 7–9

RRadon-Nikodym derivative, 14Random processes, 117Random variables, 116–117Ratchet options, 4Rayleigh quotient, 121Realized variance, 66, 72Realized volatility, 59derivatives, 65–66

Reverse Convertible Note, 10

SSABR model, 25–26Siegel’s paradox, 13, 125Spectral decomposition, 109–110Spread options, 4–5, 98Square matrix decompositions,

120–121Sticky-strike rule, 17, 18Sticky-delta rule, 17, 18Sticky-moneyness rule, 17, 18Stochastic Alpha, Beta, Rho (SABR)

model, 25–26Stochastic calculus, 117Stochastic correlation, 103–112average, 104–108

(see also Average correlation)

correlation matrices, 108–110stochastic single correlation,

103–104Stochastic volatility, 50–55.

See also Stochasticvolatility-inspired (SVI) model

connection with local volatility,53

and forward start options, 54hedging theory, 51–52and jumps, 54–55Monte Carlo simulations,

53–54problems, 55

Stochastic volatility-inspired (SVI)model, 23–25

Structured products, 9–11SVI model, 23–25

TToy model, 92Tradable average correlation, 104,

105–106

UUnattainability, 28, 112–113

VVanilla dispersion trades, 87–88,

93Vanilla options, 15–16, 31, 37, 44,

47, 50–51Variance dispersion trades, 89–91,

94Variance futures, 62

152 INDEX

Variance swaps, 60–65, 71forward variance, 64–65hedging and pricing,

62–64market, 62payoffs, 61–62variance futures, 62

Vega notional, 61VIX. See also Volatility

Index (VIX)Volatility derivatives, 59–72implied, 67–69realized, 65–66

variance swaps, 60–65volatility trading, 59–60, 70–71

Volatility Index (VIX), 67VIX futures, 67–68VIX options, 68–69

Volatility trading, 59–60, 70–71Volga, 54

WWiener process, 117Worst-of options, 6Worst-of-put pricing, 86Wright-Fisher process, 103–104

advanced equity derivatives volatility and correlations

Economy & Finance

standard brownian

monte carlo

geometric

standard brownian

month rolling

monte carlo

schmidts orthonormalization

2f sv